Aline de Oliveira Ferreira Contributions to Array Signal ... · To Dr. Marcello Campos, whose...

Aline de Oliveira Ferreira

Contributions to Array Signal Processing:Space and Space-Time Reduced-Rank

Processing and Radar-EmbeddedCommunications

Tese de Doutorado

Thesis presented to the Programa de Pos-graduacao em En-genharia Eletrica of PUC–Rio in partial fulfillment of the re-quirements for the degree of Doutor em Ciencias – EngenhariaEletrica.

Advisor : Prof. Raimundo Sampaio NetoCo–Advisor: Prof. Jose Mauro Pedro Fortes

Rio de JaneiroFebruary 2017

DBD

PUC-Rio - Certificação Digital Nº 1321805/CA


Contributions to Array Signal Processing:Space and Space-Time Reduced-Rank

Processing and Radar-EmbeddedCommunications

Thesis presented to the Programa de Pos-Graduacao em Enge-nharia Eletrica of the Departamento de Engenharia Eletrica doCentro Tecnico Cientıfico of PUC–Rio as partial fulfillment of therequirements for the degree of Doutor em Engenharia Eletrica.Approved by the undersigned Examination Committee.

Prof. Raimundo Sampaio NetoAdvisor

Centro de Estudos em Telecomunicacoes — PUC–Rio

Prof. Jose Mauro Pedro FortesCo–Advisor

Centro de Estudos em Telecomunicacoes — PUC–Rio

Prof. Marcello Luiz Rodrigues de CamposUFRJ

Prof. Jose Antonio Apolinario JuniorExercito Brasileiro

Prof. Ernesto Leite PintoIME

Fabian David BackxInstituto de Pesquisas da Marinha

Cesar Augusto Medina SotomayorIME

Prof. Marcio da Silveira CarvalhoCoordinator of the Centro Tecnico

Cientıfico da PUC–Rio

Rio de Janeiro, February 17th, 2017

DBD


All rights reserved.


Aline de Oliveira Ferreira received her undergraduation Di-ploma in Engenharia de Telecomunicacoes from the Universi-dade Federal Fluminense (UFF), Niteroi, in 2009. She earnedher Master diploma in Engenharia Eletrica from the PontifıciaUniversidade Catolica do Rio de Janeiro (PUC–Rio) in 2011.She has been with the Instituto de Pesquisas da Marinha(IPqM) as a researcher since 2010.

Bibliographic data

Ferreira, Aline de Oliveira

Contributions to array signal Processing: space and space-time reduced-rank processing and radar-embedded commu-nications / Aline de Oliveira Ferreira; advisor: RaimundoSampaio Neto; co–advisor: Jose Mauro Pedro Fortes. – 2017.

204 f.: il. color. ; 30 cm

Tese (doutorado) - Pontifıcia Universidade Catolica doRio de Janeiro, Departmento de Engenharia Eletrica, 2017.

Inclui bibliografia

1. Engenharia Eletrica – Teses. 2. Processamento espa-cial. 3. Processamento espacio-temporal. 4. Esquema dereducao de posto baseado em interpolacao e decimacao con-junta. 5. Radares de funcao dual. 6. Modulacao dos lobuloslaterais. I. Sampaio Neto, Raimundo. II. Fortes, Jose MauroPedro. III. Pontifıcia Universidade Catolica do Rio de Janeiro.Departmento de Engenharia Eletrica. IV. Tıtulo.

CDD: 621.3

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To my children, Mateus and Julieta.

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Acknowledgments

To my advisor Dr. Raimundo Sampaio Neto and co-advisor Dr. Jose

Mauro Fortes, who have accepted to advise me during a challenging period in

an area very different from their comfort zone. Their support was indispensable

for the conclusion of this wok.

To Dr. Moeness Amin, Dr. Aboulnasr Hassanien and Dr. Juan Ramirez,

who have opened the doors for me to a very interesting research topic.

To Dr. Marcello Campos, whose classes of “Array Processing” were very

inspiring.

To the Brazilian Navy (Marinha do Brasil - MB) and the Naval Research

Institute (Instituto de Pesquisas da Marinha - IPqM), especially to Edson Ne-

mer and Commanders Carla Martins and Marcelo Felzky, who have authorized

me to attend the PhD course. To PUC–Rio and CETUC for the support and

kind staff.

To my grand–parents, Lea Pessoa de Oliveira and Aluısio Osorio Pinto.

To my mother, Denise de Oliveira Pinto, and father, Afonso Celso Silva

Junior. To my husband, Diogo Teixeira Ferreira, and my children, Mateus

de Oliveira Ferreira and Julieta de Oliveira Ferreira. Their presence, support,

love, encouragement and understanding gave me the strength I needed to keep

researching.

To my IPqM colleagues, Fabian David Backx, Sergio Neves, Rodolfo Lima

and Angela Moulin. Their availability for discussing, reviewing, listening and

encouraging me was essential during the whole process of finishing this thesis.

DBD


Abstract

Ferreira, Aline de Oliveira; Sampaio Neto, Raimundo (Advisor);Fortes, Jose Mauro Pedro (Co-Advisor). Contributions to Array

Signal Processing: Space and Space-Time Reduced-Rank

Processing and Radar-Embedded Communications. Rio deJaneiro, 2017. 204p. Tese de Doutorado — Departamento de Enge-nharia Eletrica, Pontifıcia Universidade Catolica do Rio de Janeiro.

Array processing is an area with many civilian and military applications,

e.g. sonar, radar, seismology and wireless communications. By means of

space and space-time processing it is possible to enhance their features and

explore new possibilities. This area has been attracting increasingly more

attention and gathering more efforts of the science community, especially

now, that phased array antennas are established as a commercial and mature

technology. Within this context, we address the problem of reduced rank

processing in space and space-time radar signal processing and the new area

of dual-function radar-communications (DFRC), which may be summarized

as embedding communication messages into radar emissions as a secondary

task for the radar. In this thesis, we investigate the application of a new joint

interpolation and decimation rank reducing scheme in two different areas:

beamforming and space-time radar processing. This rank reducing algorithm

was never tested within these contexts before and shows impressive results.

We also propose simplifications for decreasing the computational complexity

of the algorithm in beamforming. In the topic of DFRC, we propose

two original robust radar-embedded sidelobe phase/amplitude modulation

methods which have simple closed form equations. The proposed methods

are much simpler than the state of the art and have superior performance

in terms of robustness and real-time applicability.

KeywordsBeamforming; Space-time adaptive processing (STAP); Joint interpol-

ation and decimation rank reducing scheme; Dual function radar; Sidelobe

modulation.

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Resumo

Ferreira, Aline de Oliveira; Sampaio Neto, Raimundo; Fortes, JoseMauro Pedro. Contribuicoes ao Processamento em Arranjos

de Sensores: Processamento Espacial e Espacio-Temporal

com Posto Reduzido e Radares com Comunicacoes In-

corporadas. Rio de Janeiro, 2017. 204p. Tese de Doutorado— Departamento de Engenharia Eletrica, Pontifıcia UniversidadeCatolica do Rio de Janeiro.

Processamento em arranjos de sensores e uma area com vasta aplicacao,

tanto civil quanto militar, por exemplo em sonar, radar, sismologia e comu-

nicacıes sem fio. Por meio de processamento espacial e espacio-temporal

e possıvel melhorar suas funcionalidades e explorar novas possibilidades.

Esta area vem atraindo cada vez mais a atencao e os esforcos da comunid-

ade cientıfica, especialmente agora, em que antenas phased-array se es-

tabeleceram como uma tecnologia comercial e madura. Neste contexto,

nos tratamos o problema de processamento com posto reduzido em pro-

cessamento espacial (beamforming) e espacio-temporal de sinais radar e

a nova area de radares com funcao dual de radar e comunicacıes (dual-

function radar-communications, DFRC), que pode ser resumida na incor-

poracao de mensagens de comunicacıes nas transmissıes radar como uma

tarefa secundaria. Nesta tese, nos investigamos a aplicacao de um novo es-

quema de reducao de posto baseado em interpolacao e decimacao em duas

areas distintas: processamento espacial e processamento espacio-temporal

de sinais radar. Este algoritmo para reducao de posto nunca havia sido

testado nestes ambientes antes e apresentou resultados bastante expressivos.

Nos tambem propomos simplificacıes para reduzir a complexidade computa-

cional do algoritmo em bemforming. Quanto ao topico de DFRC, nos pro-

pomos dois metodos originais para incorporar modulacao de amplitude/fase

aos lobulos laterais do diagrama de irradiacao do radar de forma robusta.

Os metodos propostos sao muito mais simples do que o estado-da-arte e ap-

resentam desempenho superior em termos de robustez e aplicabilidade em

operacıes de tempo-real. Nos ainda provemos varias outras analises, com-

paracıes e contribuicıes a esta nova area.

Palavras-chaveProcessamento espacial; Processamento espacio-temporal; Esquema de

reducao de posto baseado em interpolacao e decimacao conjunta; Radares de

funcao dual; Modulacao dos lobulos laterais.

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Contents

1 Introduction 19

2 Introduction to Beamforming 22

2.1 General RF Signal Model 22

2.2 Signal Model in Beamforming 24

2.3 Snapshot in Beamforming 27

2.4 Spatial Filtering (Beamforming) 28

3 Introduction to Space-Time Processing 31

3.1 General Radar Signal Model 31

3.2 System Model in Space-Time Processing 33

3.3 Space-Time Metrics 43Adapted Pattern 43

SIR and SIR Loss 44

Doppler Space Performance 47

Probability of Detection 48

Adaptive Matched Filter (AMF) 49

4 Reduced Rank Array Processing 52

4.1 Interpolation-and-Decimation-based Dimensionality Reduction 54

5 JIDS Dimensionality Reduction Applied to Beamforming 56

5.1 System Model 57

5.2 Rank Reduction Technique for Beamforming Application 59

5.3 JIDS Rank Reduction Technique 59An Effective Design of the Interpolation Filter Specialized for Beamforming 60

Selection of the Interpolator Length 64

JIDS Simplification for Beamforming Environment - JIDSB 66

Computational Complexity 69

5.4 Simulations 72

5.5 Conclusion 79

6 JIDS Applied to Space-Time Processing 80

6.1 System Model 80

6.2 Recast of the JIDS applied to STAP 82

6.3 Simulations 83

6.4 Conclusion 94

7 Radar and Communications 96

What is a Dual Function Radar and Communications (DFRC) Radar? 99

What’s the Idea of Sidelobe Modulation? 99

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7.1 System Model 100

8 Overview of the Sidelobe Modulation Methods 102

8.1 Sidelobe Level (SLL) Modulation 102Brief Discussion about the Existing SLL Modulation Methods 104

Symbol Error Analysis 105

Examples of a 4-ASK and a BASK SLL Modulation 108Comments on the Security against Interception of the

SLL Modulation 113

8.2 Sidelobe Phase Modulation 113Method of Phase Rotational Invariance 114

Comments on the Method of Rotational Invariance forEmbedding Sidelobe Phase Modulation 116

Bit Error Rate for the Binary Case 119Comments on the Security against Interception of the

Phase Rotational Invariance Method 122Method of Common Reference 123

Bit Error Rate for the Binary Case 125Comments on the Security against Interception of the

Common Reference Method 128

9 Proposed Radar-Embedded Sidelobe Modulation 131

9.1 Considerations About Robustness 132

9.2 Proposed Method 1 137Another Interpretation of the Proposed Technique 141

Recursive Version of the Proposed Method 1 143

9.3 Proposed Method 2 145Simplifications and Receiver Pursuit 146

Eigenvalue Analysis 148

9.4 Proposed Signalling Strategies 150Proposed Signalling Strategy for SLL Modulation 150

Proposed Signalling Strategy for Phase Modulation 151

Performance of the Non-Coherent DBPSK ReceiverWhen Two Arbitrary Symbols Are Transmitted 153

10 DFRC Simulations 155

10.1 Example of the Proposed Method 1 for Sidelobe Amplitude Modulation155Recursive Version of Method 1 157

Method 1 without the Derivative Constraint 159

Recursive Version of Method 1 without the Derivative Constraint 160

10.2 Example of the Proposed Method 2 for Sidelobe Amplitude Modulation161Method 2 without the Derivative Constraint 163

10.3 Robustness Analysis 165

10.4 Example of the Proposed Method 1 for Sidelobe Phase Modulation 167Recursive Version of Method 1 172

Method 1 without the Derivative Constraint 173

Recursive Version of Method 1 without the Derivative Constraint 174

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10.5 Example of the Proposed Method 2 for Sidelobe Phase Modulation 175Method 2 without the Derivative Constraint 177

10.6 Robustness Analysis for the Proposed Sidelobe Phase ModulationMethods 180

10.7 Computational Complexity 182

10.8 Clutter Analysis 183

10.9 Conclusion 188

11 Conclusion 190

A Appendix 199

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List of Figures

2.1 Reception process in the low-pass equivalent representation. 232.2 Coordinate system. 252.3 Possible coordinate system for a ULA. 262.4 Another possible coordinate system for a ULA. 272.5 Complete reception in the equivalent low-pass representation for

the m-th array element. 272.6 Diagram of beamforming in transmission. 292.7 Diagram of beamforming in reception. 29

3.1 Illustration of radar burst. 323.2 Platform geometry [1]. 353.3 Top view of the platform geometry [1]. 353.4 Radar CPI datacube. 393.5 Example of clutter ridges for βc = 0, βc = 0.5, βc = 1.0 and

βc = 2.5 for a PRF = 300 Hz. 423.6 Capon spectrum of R. 453.7 Normalized quiescent pattern. 453.8 Principal cuts of the normalized quiescent pattern. 453.9 normalized adapted pattern of the space-time MVDR with perfect

knowledge of the space-time covariance matrix. 463.10 Principal cuts of the normalized adapted pattern given by the

space-time MVDR with perfect knowledge of the space-timecovariance matrix at the target coordinates. 46

3.11 SIR Loss vs. Doppler frequency for quiescent space-time filter andMVDR space-time filter, fixed ϑ = 0. 48

3.12 Normalized output power range profile. 51

5.1 Block diagram of rank reduction stage followed by MVDR beam-former filter. 59

5.2 Illustration of the JIDS rank reduction stage. 605.3 SINR loss as a function of the filter lengths, Lv, for different

decimation factor, F , for an array with M = 64 elements, SNR= 10 dB and 3 jammers with JNR = -9 dB. 65

5.4 SINR loss as a function of the filter lengths, Lv, for differentdecimation factor, F , for an array with M = 64 elements, SNR= 10 dB and 4 jammers with JNR = 15 dB. 66

5.5 SINR loss as a function of the decimation factor, F , for theproposed and the optimal decimation pattern selection strategiesfor an array withM = 64 elements, SNR = 10 dB and 8 jammerswith JNR = 15 dB. 69

5.6 Comparison of the number of complex multiplications of the stageof decimation pattern selection of the JIDS and JIDSB for Lv = Fand Nit = 5. 71

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5.7 Comparison of the number of complex multiplications of theJIDSB with the CG-MWF as a function of the sample support,Ns, for F = {2, 4, 8, 16}, M = 64, Nit = 5, DMWF = 30 andNB = Ns. 73

5.8 Output SINR loss versus the sample support with M = 64, 2jammers with JNR = 15 dB and γ = σ2

n. 755.9 Output SINR loss versus the sample support with M = 64, 2

jammers with JNR = 15 dB and γ = 10σ2n. 75

5.10 Output SINR loss versus the sample support with M = 64, 2jammers with JNR = 15 dB and γ = 10σ2

n when the desiredsignal is present during estimation of the autocorrelation matrix. 76

5.11 Output SINR loss versus the sample support with M = 64, 6jammers with JNR = 15 dB and γ = σ2

n. 775.12 Output SINR loss versus the sample support with M = 64, 6

jammers with JNR = 15 dB and γ = 10σ2n. 77

5.13 Output SINR loss versus the sample support with M = 64, 6jammers with JNR = 15 dB and γ = 10σ2

n when the desiredsignal is present during estimation of the autocorrelation matrix. 78

5.14 Adapted beampattern for different algorithms for sample supportNs = 100, M = 64 and 6 jammers with JNR = 15 dB. 78

5.15 Semi-analytical BER for different algorithms for sample supportNs = 20, M = 64, 6 jammers with JNR = 15 dB. 78

6.1 Datacube showing selection of reference and guard cells forestimating the covariance matrix for a given cell under test (CUT). 81

6.2 Capon spectrum of R. 836.3 Eigenspectrum of the covariance matrix R. 846.4 SINR Loss (LSINR) vs. decimation factor for the JIDS algorithm

according the described example and with Ke = 648, ξt = 1,ϑt = 0 and ft = 100 Hz. 85

6.5 LSINR vs. Doppler with sample support of Ke = 648, fixed ϑt = 0. 866.6 LSINR vs. Doppler with sample support of Ke = 50, fixed ϑt = 0. 866.7 PD vs. PFA for Ke = 648, ξt = 1, ϑt = 0 and ft = 100 Hz, all

other parameters are the same as the former examples. 876.8 PD vs. PFA for Ke = 50, ξt = 1, ϑt = 0 and ft = 100 Hz, all

other parameters are the same as the former examples. 876.9 Pattern of the space-time MVDR-SMI with sample support of

648 - SINR = 12 dB. 886.10 Pattern of the space-time JIDS-SMI with sample support of 648

- SINR = 14 dB. 886.11 Pattern of the space-time MWF-SMI with sample support of 648

- SINR = 13 dB. 886.12 Pattern of the space-time CSM-SMI with sample support of 648

- SINR = 10 dB. 896.13 Pattern of the space-time PC-SMI with sample support of 648 -

SINR = -8 dB. 896.14 Pattern of the space-time MVDR-SMI with sample support of 50

- SINR = -7 dB. 89

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6.15 Pattern of the space-time JIDS-SMI with sample support of 50 -SINR = 13 dB. 90

6.16 Pattern of the space-time MWF-SMI with sample support of 50- SINR = -7 dB. 90

6.17 Pattern of the space-time CSM-SMI with sample support of 50 -SINR = -15 dB. 90

6.18 Pattern of the space-time PC-SMI with sample support of 50 -SINR = -27 dB. 91

6.19 Range profile for the Mountain Top data set using the unadaptedspace-time filter wQ = sst and a sample support of 20 snapshots. 92

6.20 Range profile for the Mountain Top data set using the full rankMVDR-SMI algorithm with diagonal loading and a sample supportof 20 snapshots. 93

6.21 Range profile for the Mountain Top data set using the MWFalgorithm and a sample support of 20 snapshots. 93

6.22 Range profile for the Mountain Top data set using the JIDSalgorithm and a sample support of 20 snapshots. 93

6.23 Range profile for the Mountain Top data set using the PC-SMIalgorithm and a sample support of 20 snapshots. 94

6.24 Range profile for the Mountain Top data set using the CSM-SMIalgorithm and a sample support of 20 snapshots. 94

7.1 Transmit power pattern generated by the transmit beamformerw0, which embeds symbol “0” towards the receiver direction. 100

7.2 Transmit power pattern generated by the transmit beamformerw1, which embeds symbol “1” towards the receiver direction. 100

7.3 Coordinate system. 101

8.1 Illustration of the non-coherent ASK receiver. 1068.2 Illustration of decision regions for a 4-ASK constellation. 1078.3 Power patterns with embedded amplitude modulation at uc =

−0.6 using (8.4) - (8.6). 1098.4 Conditional probability density function of the symbols transmit-

ted using a 4-ASK constellation embedded using the beamformersof Fig. 8.3 and SNR = 10 dB. 109

8.5 Conditional probability density function of the symbols transmit-ted using a 4-ASK constellation embedded using the beamformersof Fig. 8.3 and SNR = 20 dB. 110

8.6 Conditional probability density function of the symbols transmit-ted using a 4-ASK constellation embedded using the beamformersof Fig. 8.3 and SNR = 5 dB. 110

8.7 Beamformers chosen to embed a binary amplitude constellationtowards uc = −0.6. 111

8.8 Conditional probability density function of the symbols transmit-ted using a BASK constellation embedded using the beamformersof Fig. 8.7 and SNR = 5 dB. 111


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8.11 Symbol error probability for the BASK and 4-ASK constellationvs. SNR (dB). 113

8.12 Angular BER for a binary amplitude modulation using the methodof convex optimization described in [2]. 114

8.13 Illustration of the signalling method of [3]. 1158.14 Principal beampattern. 1168.15 Phase towards u = 0.6428 generated by all the 512 beamformers. 1178.16 Phase difference towards u = 0.6428 generated by 262144

beamformer pairs. 1178.17 Phase towards u = 0.5 generated by all the 512 beamformers. 1188.18 Phase difference towards u = 0.5 generated by 262144 beam-

former pairs. 1188.19 Illustration of the rotational invariance-based non-coherent PSK

receiver. 1208.20 Transmit power pattern generated by the beamformers w0 and w0.1218.21 Transmit power pattern generated by the beamformers w1 and w1.1218.22 Phase difference towards uc = 0.6428 of the beamformer pairs

(w1,w1) and (w2,w2) in polar coordinates. 1218.23 Bit error probability for the non-coherent BPSK vs. SNR (dB)

using the phase rotational invariance method. 1228.24 Phase difference in the u-space generated by the beamformer

pairs (w1, w1) and (w2, w2). 1238.25 Angular BER for a BPSK modulation using the method of

rotational invariance described in [3]. 1238.26 Illustration of the signalling method of [4] for the sequence of

symbols φ1, φ2 and φ3. 1258.27 Illustration of the common reference-based non-coherent PSK

receiver. 1278.28 Set of beamformers, w0, w1 and w2, used to embed a BPSK

constellation, φ1 = 0 and φ2 = π, using the signalling strategy of[4]. 127

8.29 Phase difference towards uc of the beamformer pairs (w1,w0) and(w2,w0). 128

8.30 Bit error probability for the non-coherent BPSK vs. SNR (dB)using the common reference method. 129

8.31 Phase difference of the beamformer pairs (w1,w0) and (w2,w0)in the u-space. 130

8.32 BER in the u-space for the described BPSK system using thebeamformer pairs (w1,w0) and (w2,w0). 130

9.1 Maximum range as a function of SNRmin for the first example. 1359.2 Moving communication receiver. 1369.3 Zoom of BASK angular BER showing the BER degradation

according to communication receiver motion for the first examplefor a SNR of 15 dB. 136

9.4 Maximum range as a function of SNRmin for the second example. 137

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9.5 Zoom of non-coherent BPSK angular BER showing the BERdegradation according to communication receiver motion for thesecond example for a SNR of 10 dB. 138

9.6 Eigenspectrum of Q for different values of M . 1499.7 Number of eigenvalues necessary to comprise 95% of the sum of

all eigenvalues vs. M , for several fidelity regions. 1499.8 Illustration of the non-coherent ASK receiver. 1509.9 Diagram of the equivalent low-pass receiver for DPSK modulated

signals. 1529.10 Illustration of a D8PSK modulation and its decision regions. 153

10.1 Beamformers that embed a binary amplitude constellation to-wards uc = −0.6 using the method of [5]. 156

10.2 Power pattern of beamformers w1 and w2 that embed a binaryamplitude constellation towards uc = −0.6 generated using theproposed Method 1 for α = 0.3. 156

10.3 Mainbeam and total rms error vs. α due to beamformers w1(α)and w2(α) depicted in Fig. 10.2 for α = 0.3. 157

10.4 Power pattern of beamformers w1,r and w2,r that embed a binaryamplitude constellation towards uc = −0.6 generated using therecursive version of the proposed Method 1 for α = 0.3. 158

10.5 Squared error vs. number of iterations of the beamformers w1,r

and w2,r, depicted in Fig. 10.4, generated using the recursiveversion of the proposed Method 1 for α = 0.3. 159

10.6 Beamformers that embed a binary amplitude constellation to-wards uc = −0.6 using the proposed Method 1 for α = 0.5without the first derivative null constraint. 160

10.7 Mainbeam and total rms error vs. α for the case without the firstderivative null constraint due to beamformers w1(α) and w2(α)depicted in Fig. 10.6. 160

10.8 Power pattern of beamformers w1,r and w2,r that embed a binaryamplitude constellation towards uc = −0.6 generated using therecursive version of the proposed Method 1 for α = 0.5 for thecase without the first derivative null constraint. 161


and w2,r, depicted in Fig. 10.8, generated using the recursiveversion of the proposed Method 1 for α = 0.5 for the casewithout the first derivative null constraint. 161

10.10 Power pattern of beamformers w1,e and w2,e that embed a binaryamplitude constellation towards uc = −0.6 generated using theproposed Method 2 for Ne = 5. 162

10.11 Mainbeam and total rms error vs. Ne due to beamformersw1,e(Ne) and w2,e(Ne) depicted in Fig. 10.10. 163

10.12 Beamformers, w1,e and w2,e, that embed a binary amplitudeconstellation towards uc = −0.6 using the proposed Method 2for Ne = 5 without the first derivative null constraint. 164

10.13 Mainbeam and total rms error vs. Ne for the case without thefirst derivative null constraint due to beamformers w1,e(Ne) andw2,e(Ne) depicted in Fig. 10.12. 164

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10.14 Conditional probability density function of the symbols trans-mitted using the binary constellation C = {C1, C2} =

{√10−20/10,

√10−40/10} and SNR = 15 dB. 165

10.15 BER × u, for embedded amplitude modulation at uc = −0.6, forthe proposed Method 1 with and without the derivative constraintand the method of [5]. 166

10.16 BER × u, for embedded amplitude modulation at uc = −0.6, forthe proposed Method 2 with and without the derivative constraintand the method of [5]. 166

10.17 Zoom of the BER × u, for embedded amplitude modulation atuc = −0.6, for the proposed Method 1 with and without thederivative constraint and the method of [5]. 167

10.18 Beamformers that embed a DBPSK constellation towards uc =−0.6 using the method of [4]. 168

10.19 Phase difference of the beamformer pairs of Fig. 10.18 (w1,w0)and (w2,w0) in the u-space. 169

10.20 Power pattern of beamformers w1 and w2 that embed a DBPSKmodulation towards uc = −0.6 generated using the proposedMethod 1 for α = 0.6. 170

10.21 Phase difference of the beamformer pair of Fig. 10.20 (w2,w1)in the u-space. 170

10.22 Mainbeam and total rms error vs. α due to beamformers w1(α)and w2(α) depicted in Fig. 10.20. 171

10.23 Simulated and analytical BER for a DBPSK system. 17110.24 Power pattern of beamformers w1,r and w2,r that embed a binary

DPSK constellation towards uc = −0.6 generated using therecursive version of the proposed Method 1 for α = 0.6. 172


and w2,r, depicted in Fig. 10.24, generated using the recursiveversion of the proposed Method 1 for α = 0.6. 172

10.26 Beamformers that embed a binary DPSK constellation towardsuc = −0.6 using the proposed Method 1 for α = 0.1 withoutthe first derivative null constraint. 173


10.28 Mainbeam and total rms error vs. α for the case without the firstderivative null constraint due to beamformers w1(α) and w2(α)depicted in Fig. 10.26. 174

10.29 Power pattern of beamformers w1,r and w2,r that embed a binaryDPSK constellation towards uc = −0.6 generated using therecursive version of the proposed method 1 for α = 0.1 for thecase without the first derivative null constraint. 175


and w2,r, depicted in Fig. 10.29, generated using the recursiveversion of the proposed Method 1 for α = 0.1 for the casewithout the first derivative null constraint. 175

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10.31 Power pattern of beamformers w1,e and w2,e that embed a binaryDPSK constellation towards uc = −0.6 generated using theproposed Method 2 for Ne = 5. 176

10.32 Phase difference of the beamformer pair of Fig. 10.31 (w2,e,w1,e)in the u-space. 176

10.33 Mainbeam and total rms error vs. Ne due to beamformersw1,e(Ne) and w2,e(Ne) depicted in Fig. 10.31. 177

10.34 Beamformers, w1,e and w2,e, that embed a binary DPSK con-stellation towards uc = −0.6 using the proposed Method 2 forNe = 5 without the first derivative null constraint. 178

10.35 Beamformers, w1,e and w2,e, that embed a binary DPSK con-stellation towards uc = −0.6 using the proposed Method 2 forNe = 3 without the first derivative null constraint. 178


10.37 Mainbeam and total rms error vs. Ne for the case without thefirst derivative null constraint due to beamformers w1,e(Ne) andw2,e(Ne) depicted in Fig. 10.35. 179

10.38 Analytical and simulated BER × u, for embedded DBPSKmodulation at uc = −0.6, for the proposed Method 1 withoutthe derivative constraint for SNR = 5 dB. 180

10.39 BER × u, for embedded DBPSK modulation at uc = −0.6, forthe proposed Method 1 with and without the derivative constraintand the method of [4] for SNR = 10 dB. 181

10.40 BER × u, for embedded DBPSK modulation at uc = −0.6, forthe proposed Method 2 with and without the derivative constraintand the method of [4] for SNR = 10 dB. 181

10.41 Zoom of the BER × u, for embedded DBPSK modulation atuc = −0.6, for the proposed Method 1 with and without thederivative constraint and the method of [4] for SNR = 10 dB. 182

10.42 Comparison of the number of complex products and additionsnecessary for Method 1, recursive version of Method 1 andMethod 2, given α and Ne. 183

10.43 Beampatterns adopted for clutter analysis simulation. 18610.44 Original Capon spectrum of the clutter plus noise environment. 18610.45 Capon spectrum of the clutter plus noise environment using

sidelobe modulation. 18710.46 Cut of the Capon spectrum depicted in Figs 10.45 and 10.44 for

u = −0.6. 18710.47 Cut of the Capon spectrum depicted in Figs 10.45 and 10.44 for

fD = 0 Hz. 188

A.1 Illustration of a DBPSK modulation and its decision regions. 200

DBD


List of Tables

5.1 Complex operations of the JIDS. 705.2 Complex operations of the JIDSB. 705.3 JIDSB-SMI Strategy 715.4 CG-MWF Strategy 725.5 Complex products of the JIDSB-SMI and CG-MWF algorithms. 725.6 Complex additions of the JIDSB-SMI and CG-MWF algorithms. 72

9.1 Complex operations of the proposed method 1 with the derivativenull constraint. 142

9.2 Complex operations of the recursive version of the proposed method1 with the derivative null constraint. 145

9.3 Complex operations of the proposed method 2. 148

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1Introduction

This thesis is divided into two main topics related to the area of

array signal processing. The first topic addresses the problem of reduced

rank processing in space and space-time radar signal processing. The second

topic addresses the dual-function radar-communications (DFRC) problem,

which may be summarized as embedding communication messages into radar

emissions as a secondary task for the radar.

As both topics deal with space and/or space-time processing, we make

an introduction to space processing, commonly known as beamforming and

to space-time processing, commonly named space-time adaptive processing

(STAP). Chapter 2 gives the fundamentals of beamforming. We present the

general radio frequency (RF) signal model and derive the beamforming signal

model that is used throughout this thesis. We define important concepts in

beamforming like snapshot and beampattern for example. Chapter 3 makes

an introduction to space-time processing specialized for radar applications.

From the general radar model we derive the system model used in STAP and

explain the most important figures of merit and metrics used to evaluate the

performance of space-time filters. These two chapters compose the base for

understanding the rest of this thesis.

Chapter 4 introduces the reduced rank array processing topic. We give

the motivation for reducing the rank in array signal processing, for both

space only and radar (space-time) applications. We review the most relevant

rank reducing techniques existent in the literature and we introduce the rank

reduction technique based on an interpolation and decimation scheme, which

is the object of our study in this rank reducing topic. We name this method

the JIDS (joint interpolation and decimation scheme) and we present its

development history.

In Chapter 5, we specialize the JIDS for beamforming applications.

Considering this new scenario, we propose simplifications of the method,

which lead to a significantly reduction of its overall complexity. The new

specialized and simplified JIDS for beamforming is named here JIDSB. In Sec-

tion 5.4 we compare the JIDSB performance results with other renowned rank

reducing techniques. Results of this technique in a beamforming environment

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Chapter 1. Introduction 20

are new in the literature and they reveal excellent SINR loss performance with

superior robustness with a low computational complexity. The conclusions of

the rank reduced beamforming topic is presented in Section 5.5. The study on

this topic, carried out during this thesis, resulted in a paper that was submitted

to the IEEE TAES journal.

In Chapter 6, we specialize the JIDS rank reduction technique for

space-time applications, more specifically, airborne phased-array radars. In

Section 6.3 we present computer simulations and compare the performance

results of the JIDS with other established rank reducing techniques. The JIDS

has an impressive ability to significantly reduce the length of the space-time

snapshots and achieves very good performance in Doppler SINR loss and

probability of detection, especially in sample starving scenarios. These results

are very promising and new within the radar literature. The conclusions of the

rank reduced space-time processing topic is given in Section 6.4. The study on

this topic carried out during this thesis preparation resulted in the conference

paper [6].

Chapter 7 introduces the DFRC topic. We give the motivation for

coupling radar and communications and make an overview of the current

methods that address this problem, separating them in subareas according

to their specificities. Our work is inserted in the DFRC based on sidelobe

modulation subarea, which encompasses both amplitude and phase modulation

of the sidelobe of the radar transmit beampattern. We also describe how the

DFRC topic is structured within this thesis and we detail our contributions to

the topic.

Chapter 8 explains in detail the main sidelobe modulation methods

present in the literature with added simulation and analysis. We focus on

the main idea of the existent methods and we highlight their pros and cons, so

that the reader can place our contributions among the existent work. Chapter

9 thoroughly explains our contributions to the topic, which can be summarized

as:

– we propose two original sidelobe modulation methods with the

exclusive features of

– being applicable to both amplitude and phase modulation, (there

is no need of redesigning of the optimization formulation);

– being robust against small angular errors of the relative position

between the radar and the communication receiver (to the best

of our knowledge, there is no method in the literature with this

advantage) and

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Chapter 1. Introduction 21

– having low computational complexity (we derive closed form solu-

tions and we simplify them as possible);

– we derive update equations for keeping the communication link when

there is relative movement between the radar and the communication

receiver, (this subject is mentioned in the literature, but no other method

is simple enough for effectively dealing with this issue);

– we derive a recursive version of one of the proposed methods, allowing

the investigation of interesting stop criteria;

– we propose a non-coherent signalling strategy for sidelobe phase

modulation which is much simpler than the others found in the DFRC

literature;

– we derive an analytical bit error rate (BER) expression necessary for

the DFRC sidelobe phase modulation performance evaluation;

– we present the first (and so far the only) analysis of the effect of sidelobe

modulation in clutter mitigation techniques.

Chapter 10 shows several computer simulations of the proposed tech-

niques and compares the results of the proposed methods with the existent

ones. In Section 10.9 of this chapter we summarize our conclusions related to

the DFRC topic. Part of the studies relative to this topic carried out during

this thesis preparation led to papers [7] and [8].

Chapter 11 reviews some of the conclusions given throughout this thesis,

presents possibilities to build on, insights about the studied fields and possib-

ilities to explore.

Notation: the superscripts T and H denote transpose and conjugate

transpose respectively, ⊙ denotes the Hadamard matrix product, ⊗ represents

the Kronecker product, E{·} stands for the expected value and ℜ{·} returns

the real part of the argument.

DBD


2Introduction to Beamforming

Beamforming or spatial filtering is a signal processing technique used in

sensor arrays to provide a versatile directional signal transmission or reception.

This is achieved by combining the spatial samples of the propagating wave

fields in such a way that signals at particular angles (or more correctly

wavenumbers) experience constructive interference while others experience

destructive interference. This combination is accomplished by the beamformer.

Beamforming can be applied for both transmitting and receiving purposes.

Beamforming can be used for electromagnetic or sound waves. It has

found numerous applications in radar, sonar, seismology, wireless communica-

tions, radio astronomy, acoustics, and biomedicine. Adaptive beamforming is

used, for example, to detect and estimate the signal-of-interest at the output

of a sensor array by means of optimal (e.g., least-squares) spatial filtering and

interference rejection.

In this chapter we will describe the beamforming signal model and define

some fundamentals concepts in beamforming.

2.1 General RF Signal Model

Consider a real-valued RF signal, x(t), of the form

x(t) =√2Eg(t) cos(2πfct+ ρ(t) + ψ), (2.1)

where E is a constant that accounts for the signal energy, fc is the RF radar

carrier frequency, g(t) is the low-pass transmitted pulse envelope of duration

TP , ρ(t) represents the phase or frequency modulation and ψ is the initial

phase.

The real-valued signal described in (2.1) can be written in its complex,

low-pass, equivalent representation

x(t) = ℜ{x(t)ej2πfct

}, (2.2)

where the complex envelope of x(t) with respect to fc is given by

x(t) =√2Eg(t)ejψejρ(t). (2.3)

The product

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Chapter 2. Introduction to Beamforming 23

ℜ{·}

ℑ{·}

X

r(t) = x(t) + n(t)

e−jψ√2

h(t)

z1(t) zo(t)

Figure 2.1: Reception process in the low-pass equivalent representation.

s(t) = g(t)ejρ(t), (2.4)

in (2.3) has bandwidth W , W << fc, is normalized to unitary energy,

∫ TP

0

|s(t)|2dt = 1, (2.5)

and is considered to be the waveform within this thesis.

The general demodulation scheme of the received signal can be further

improved in terms of maximizing the signal to noise ratio (SNR) by the use of

a matched filter, h(t), as the reception filter. The matched filter, h(t), is given

by

H(f) = S∗(f)e−j2πfTP or (2.6)

h(t) = s∗(TP − t). (2.7)

The complete reception process in its low-complex equivalent represent-

ation is depicted in Fig. 2.1.

In Fig. 2.1, the input of the receiver is r(t), given by

r(t) = x(t) + n(t), (2.8)

=√2Es(t)ejψ + n(t), (2.9)

where s(t) is defined in (2.4) and n(t) is the low-pass equivalent AWGN

(Additive White Gaussian Noise) process, which has a flat power spectrum

Sn(f) = 2N0, for f > −fc [9]. The input after the first stage of the receiver,

z1(t), is given byz1(t) =

√Es(t) + n1(t), (2.10)

where n1(t) has a flat power spectrum Sn1(f) = N0, for f > −fc. In Fig. 2.1,

the output of the matched filter, zo(t), is composed by

zo(t) = y(t) + no(t), (2.11)

DBD



where the portion of the desired signal, y(t), is given by

y(t) =√E

∫ ∞

−∞s(α)s∗(α + Tp − t)dα, (2.12)

=√ERss(Tp − t), (2.13)

whereRss(t) =

∫ ∞

−∞s(α)s∗(α+ t)dα (2.14)

is the autocorrelation function of s(t). The noise no(t) has a power spectral

density Sno(f) = N0|H(f)|2 and variance E[|no(t)|2] = N0

∫∞−∞ |H(f)|2df =

N0

∫∞−∞ |h(t)|2dt = N0.

2.2 Signal Model in Beamforming

In this section we describe the particularities of the signal transmitted (or

received) at a generic sensor array. Now, suppose that the signal of equation

(2.1) is transmitted (or received) at the origin of the Cartesian coordinate

system (depicted in Fig. 2.2). The signal transmitted (or received) at an

arbitrary m-th element of a sensor array withM elements displaced arbitrarily

in the R3 space, is given by

x(t,dm) = ℜ{x(t,dm)e

j2πfct}, (2.15)

where dm is the position vector, given by

dm =

dx,m

dy,m

dz,m

, (2.16)

where {dx/m, dy/m, dz/m} are the Cartesian coordinates of the m-th element

phase center. The complex envelope x(t,dm) is given by

x(t,dm) =√2Es(t− τm)e

jψe−j2πfcτm , (2.17)

where s(t) is defined in equation (2.4). The time-delay between transmission

(or reception) of the plane wave at the reference point (origin of the coordinate

system) and the m-th channel, τm, is computed from the geometry given in

Fig. 2.2 as follows [10]

τm =k(φ, θ)Tdm

c, (2.18)

where c is the velocity of the propagating wave and k(φ, θ) is a unit vector

normal to the plane wave, given by

DBD



Figure 2.2: Coordinate system.

k(φ, θ) =

sin(θ) cos(φ)

sin(θ) sin(φ)

cos(θ)

, (2.19)

where φ and θ represent azimuth and elevation angles respectively.

If the narrowband assumption holds, which is the bandwidth of s(t), W ,

being much smaller than the carrier frequency, fc, and if we place the center

of the coordinate system somewhere close to the sensor array, then we can use

the following approximation

s(t− τm) ∼= s(t). (2.20)

Substituting (2.20) into (2.17), we have that the complex envelope at the m-th

element isx(t,dm) =

√2Es(t)ejψe−j2πfcτm . (2.21)

The spatial dependence on the element can be separated as

x(t,dm) = x(t)sm, (2.22)

wherex(t) =

√2Es(t)ejψ (2.23)

andsm = e−j2πfcτm . (2.24)

The M complex envelopes, x(t), due to the transmission of the M

elements can be compactly described in a vector form as

x(t) = x(t)s, (2.25)

DBD



Figure 2.3: Possible coordinate system for a ULA.

where s is the steering vector (also known as the array manifold) given by

stacking sm, m = 0, . . . ,M − 1, in a vector form,

s =

e−j2πfcτ0

e−j2πfcτ1

...

e−j2πfcτM−1

. (2.26)

Depending on where we place the coordinate system relative to the array

we can achieve different forms of writing the steering vector. A given form may

be more convenient than others depending on the application. For example,

let d be the uniform sub-array spacing of an M channel uniform linear array

(ULA). Also, place the first sensor at the origin and designate it as the phase

reference, define θ as the angle formed by the normal to the planewave and

the normal to the ULA as illustrated in Figure 2.3. Thus, the visible region is

θ ∈ [−90o, 90o] and the steering vector is given by

s =

1

e−j2πdλc

sin(θ)

e−j2π2dλc

sin θ)

...

e−j2π(M−1) dλc

sin(θ)

. (2.27)

Now, place the center of the coordinate system at the center of a M

element ULA and define θ as the angle formed by the normal to the plane

wave and the line of the ULA, as depicted in Fig. 2.4. Thus, the visible region

corresponds to the interval θ ∈ [0o, 180o] and the steering vector, s, can be

written as

s =

ej(−M−1

2 ) 2πdλc

cos(θ)

ej(1−M−1

2 ) 2πdλc

cos(θ)

...

ej(M−1−M−12 ) 2πd

λccos(θ)

. (2.28)

DBD



Figure 2.4: Another possible coordinate system for a ULA.

X h(t)

r(t,dm) r[i] = zo(iTs)

e−jψ√2

t = iTsz1(t,dm) zo(t,dm)

Figure 2.5: Complete reception in the equivalent low-pass representation forthe m-th array element.

2.3 Snapshot in Beamforming

The reception happens at each element, therefore, for them-th element at

position dm, the received signal, in the complex equivalent form, after matched

filtering, zo(t,dm), is given by

zo(t,dm) = y(t,dm) + nm(t), (2.29)

where nm(t) is the AWGN low-pass equivalent noise process with flat power

spectrum Snm , given by Snm = N0 within the frequencies limited by the

matched filter and y(t,dm) is given by

y(t,dm) = y(t)sm, (2.30)

where y(t) is described in (2.12) and sm, defined in (2.24), is the term which

contains the spatial dependence

sm = e−j2πfcτm .

The signal is then sampled in a rate of Rs = 1/Ts, where Ts is the

sampling interval. This process is depicted in Fig. 2.5. The output of the

sampling process results in samples given by r[i], as

rm[i] = zo(t,dm)|t=iTs, (2.31)

= y(iTs)sm + nm(iTs). (2.32)

DBD



The snapshot stacks the information of all sensor elements at the same

sampling instant, like a photograph of the output of all sensors at a given

instant. The snapshot is, thus, given by

r[i] = y(iTs)s + no(i), (2.33)

where s is the spatial steering vector defined in (2.26) and no(i) =

[n0(iTs), n1(iTs), . . . , nM−1(iTs)]T is the noise vector.

2.4 Spatial Filtering (Beamforming)

When all sensor elements are able to add independently a gain and

a phase shift to their transmitted (or received) signals, we say that the

sensor array is able to beamform. The joint operation of individual sensors

modifying amplitude and phase of the transmitted (or received) signals is called

beamforming. Due to the composition of the gain and phase-shifts with the

array delay of all elements, the amplitude towards each direction within the

visible region doesn’t necessarily follow the omnidirectional radiation pattern,

though the individual sensors are usually omnidirectional. The result of this

joint operation allows the sensor array to perform spatial filtering.

The transmitted (or received) signal, composed of the individual con-

tribution of all sensor elements, depends on the gain and phase shift of each

sensor and depends on the intrinsic array delay, τm. The delay, τm, depends

on both the array geometry and the direction of the transmitted plane wave.

Given the array geometry and the plane wave direction, one can define the

steering vector, s. Having defined the steering vector for a specific direction,

θ, s(θ), we can visualize the beamforming effect over the visible region.

The beamforming process for transmission in a M element array is

depicted in Fig. 2.6. The transmitted signal, x(t, θ), irradiated towards θ due

to the contribution of all elements of the sensor array is given by

x(t, θ) = wHs(θ)x(t), (2.34)

where x(t) is the complex envelope of x(t) with respect to fc defined in (2.3)

and w = [w0, . . . , wM−1]T ∈ CM×1 is the complex weighting vector with the

weights of each branch of the array.

The beamforming process for reception is depicted in Fig. 2.7. As we

multiply by the beamformer weight, which is equivalently to adding a gain

and a phase-shift to the individual sensor branches, the received signal will

also be modified by the overall composition of gain, phase-shifts and array

delay. The received sample due to the contribution of all sensors, rB[i], as

DBD



w∗

0

w∗

1

w∗

M−1

...

x(t,d1) = x(t)s1

x(t,dM−1) = x(t)sM−1

xB(t,d0) = x(t)w∗

0s0

xB(t,dM−1) = x(t)w∗

M−1sM−1

x(t,d0) = x(t)s0ℜ{·}

xB(t,d0) = x(t)w∗

1s1ℜ{·}

ℜ{·}

e2πfct

e2πfct

e2πfct

x

x

x

Figure 2.6: Diagram of beamforming in transmission.

w∗

0

w∗

1

w∗

M−1

+

t = iTs

rB [i]

...

zo(t,dM−1) = y(t)sM−1

zB(t,d1) = y(t)w∗

1s1

zB(t,dM−1) = y(t)w∗

M−1sM−1

zo(t,d0) = y(t)s0 + n0(t)

zo(t,d1) = y(t)s1 + n1(t)

+nM−1(t) +w∗

M−1nM−1(t)

+w∗

1n1(t)

zB(t,d0) = y(t)w∗

0s0+w∗

0n0(t)

rB(t)

Figure 2.7: Diagram of beamforming in reception.

depicted in Fig. 2.7, can be written as

rB[i] = wHr[i], (2.35)

with r[i] defined in (2.33). Thus,

rB[i] = y(iTs)wHs(θ) +wHno. (2.36)

We define the beampattern, B(θ), as

B(θ) , wHs(θ) (2.37)

DBD



and the normalized beampattern, B(θ), as

B(θ) ,B(θ)

B(θo), (2.38)

where θo is the direction towards which the beampattern, B(θ), is maximum.

If we vary θ within the visible region we can visualize the radiation power

pattern of the array.

The beampattern allows us to visualize the effect of constructive and

destructive interference within the visible region in space.

Many methods exist for defining the weights, wm, in order to emphasize

the signals impinging from a desired direction and to cancel signals impinging

from other directions.

DBD


3Introduction to Space-Time Processing

So far, in Chapter 2, we have derived the signal model for spatial

filtering (or beamforming). The understanding of this model is necessary for

understanding the model of radars which have the transmitting or receiving

parts composed of an array of sensors. But in radar applications, targets

are commonly moving. It implies that the echoes from a target moving

relative to the radar with a radial velocity of v m/s will have a Doppler

shift proportional to its velocity. Exploiting the angle (spatial) dependency

and Doppler (temporal) dependency, one can achieve better results in terms

of radar signal processing. In this chapter we will derive the signal model and

explain the common metrics of space-time processing, or space-time adaptive

processing (STAP), as it is usually known in the literature.

The most fundamental problem in radar signal detection is to uncover an

object or physical phenomenon against a background of interference consisting

of clutter (echoes from the environment), one or more jammers (intentional

interference) and background noise. This requires determining whether the

receiver output at a given time (or cell under test - CUT - which is associated

to an specific range gate) represents the echo from a reflecting target or only

noise. Detection decisions are usually made by comparison of some statistic

test to a threshold.

By dealing simultaneously with both domains, spatial and temporal, we

can achieve more degrees of freedom (DoF). Spatial and temporal signal DoF

greatly enhance radar detection rate when the target competes with ground

clutter (echoes from the environment at the ground level) and barrage noise

jamming [11]. Ground clutter returns exhibit correlation in both spatial and

temporal dimensions, while jamming is predominantly correlated in angle

for modest bandwidth. The Doppler-wavenumber or angle-Doppler spectrum

provides a unique representation of a signal in a three dimensional plane.

3.1 General Radar Signal Model

The low-pass complex envelope of a typical RF pulsed radar burst

composed of J pulses is given by

DBD


Chapter 3. Introduction to Space-Time Processing 32

Time

Am

plitu

de

0

τ

Pulse 0 Pulse 1 Pulse 2 Pulse 3 Pulse 4

CPI

TPRI

s(t)

Figure 3.1: Illustration of radar burst.

xB(t) =√

2Etejψ

J−1∑

k=0

s(t− kTPRI), 0 ≤ t < JTPRI , (3.1)

where Et is the RF pulse energy, ψ is the initial carrier phase, TPRI is the

pulse repetition interval (PRI) and s(t) is the radar waveform, which is a low-

pass pulse of duration τ , with bandwidth W << fc, where fc is the carrier

frequency, normalized to unitary energy,∫ τ

0

|s(t)|2dt = 1. (3.2)

The radar waveform, s(t), is usually of the form

s(t) = g(t)ejρ(t), (3.3)

where g(t) is the low-pass transmitted pulse envelope of duration τ and ρ(t)

represents the phase or frequency modulation.

In Fig. 3.1 we illustrate a burst of five (J=5) simple pulses of duration

τ . The total duration (JTPRI) is the coherent processing interval (CPI).

The time required for a pulse to propagate a distance R and return,

travelling a total distance 2R, is 2R/c, where c is the speed of light. Thus, if at

delay time t0, the received amplitude of the radar echo is greater than a given

threshold, it is assumed that a target is present at range

R =ct02. (3.4)

The maximum unambiguous range, Rmax, is related to the radar PRI, TPRI ,

by

DBD



Rmax =cTPRI2

. (3.5)

In order to maximize the detection range, most radar systems try to maximize

the transmitted energy. One way to do this is to always operate the transmitter

at full power during a pulse. That is why radars generally do not use amplitude

modulation. The longer the pulse, the higher is the average energy as well.

On the other hand, the nominal range resolution ∆R is determined by the

waveform bandwidth W , as∆R =

c

2W. (3.6)

For an unmodulated pulse, the bandwidth is inversely proportional to its

duration, what would lead to a resolution of ∆R = cτ2. To increase the

waveform bandwidth without reducing its pulse length and therefore without

sacrificing energy, many radars routinely use phase or frequency modulation.

Two scatterers are resolved if they produce two separately identifiable signals

at the system output, as opposed to combining into a single undifferentiated

output.

Desirable values of range resolution vary from a few kilometers in long-

range surveillance systems, which tend to operate at lower RFs, to a meter

or less in very fine resolution imaging systems, which tend to operate at

high RFs. Corresponding waveform bandwidths are on the order of 100

KHz to 1 GHz, and are typically 1 percent or less of the RF. Few radars

achieve 10 percent bandwidth. Thus, most radar waveforms can be considered

narrowband bandpass functions [12].

3.2 System Model in Space-Time Processing

We consider a pulsed-Doppler radar with K radiating elements placed in

a uniform linear array (ULA), for simplicity of calculations, but the array

can have an arbitrary shape to which the system model can be adapted.

All elements are assumed to have the same radiation pattern. In order to

perform pulse integration [12], J pulses are transmitted within a CPI. Pulses

are transmitted in a pulse repetition frequency of 1/TPRI .

The analytical signal of the echo waveform, rk(t), from a target modelled

as a single scatterer, received by the k-th element, ignoring relativistic effects,

is given byrk(t) = ars(t− τe)e

j2π(fc+fD)(t−τe)ejψ, (3.7)

where ar is the echo amplitude and

fD =2vtλc

(3.8)

is the target Doppler frequency, vt is the relative radial target velocity with

respect to the radar and λc is the carrier wavelength. The target delay is given

DBD



byτe = τRT + τk, (3.9)

where τRT = 2Rt/c is the target round-trip time measured at the phase

reference, Rt is the distance to the target and c is the speed of light. If the target

is moving then Rt is actually a function of the time, Rt(t), but for instantaneous

velocities that are a small fraction of the speed of light, the commonly quasi-

stationary assumption is made [12], so that Rt(t) ∼= Rt. Variable τk is the

relative delay measured from the phase reference to the k-th element.

The received analytical signal, rk(t), is thus

rk(t) = ars(t− τRT − τk)ej2π(fc+fD)(t−τRT−τk)ejψ, (3.10)

= arejψs(t− τRT − τk) ·

· ej2πfcte−j2π(fc+fD)τRT e−j2π(fc+fD)τkej2πfDt.

If the narrowband assumption is valid, which is that the bandwidth of

s(t), W , is much smaller than the carrier frequency, fc, if we place the center

of the coordinate system somewhere close to the sensor array and assuming

that fD << fc, we can make the following approximations,

s(t− τRT − τk) ∼= s(t− τRT ), (3.11)

e−j2π(fc+fD)τk ∼= e−j2πfcτk . (3.12)

The analytical signal rk(t) is approximated to

rk(t) ∼= arejψ′

s(t− τRT )ej2πfcte−j2πfcτkej2πfDt, (3.13)

where ejψ′= ejψe−j2π(fc+fD)τRT .

The coordinate system assumed here is depicted in Figs. 3.2, which is the

same as the one described in [1]. The variables φ and θ refer to true azimuth

and elevation respectively, and not the standard spherical coordinate system

angles.

A unit vector k(φ, θ), pointing in the (φ, θ) direction according to this

coordinate system is given by

k(φ, θ) =

cos(θ) sin(φ)

cos(θ) cos(φ)

sin(θ)

. (3.14)

In our work we consider that the array is located at z = 0, as depicted in Fig.

3.3. The k-th, k = 0, . . . , K − 1, element of the ULA is located at position

DBD



Figure 3.2: Platform geometry [1].

Figure 3.3: Top view of the platform geometry [1].

dk =

kd

0

0

, (3.15)

where d is the ULA inter-element. Thus, the time-delay τk is expressed as

τk =kTdk

c= k

d

ccos(θt) sin(φt), (3.16)

where (θt, φt) are the elevation and azimuth angles of the target, and c is the

speed of light. The quantity, ϑ, the target spatial frequency, is conveniently

defined as

DBD



ϑt =d

λccos(θt) sin(φt), (3.17)

where, λc is the carrier wavelength, so that

2πfcτk = 2πkϑt. (3.18)

Assuming an ideal coherent receiver, the analytical complex downcon-

verted received signal is given by

rk(t) = rk(t)e−j2πfct√

2, (3.19)

rk(t) = σts(t− τRT )ej2πkϑtej2πfDt, (3.20)

where σt =arejψe−j2π(fc+fD)τRT√

2. In each element, rk(t), is match filtered with

the receiver filter, h(t) = s∗(τ − t). The output of the k-th channel, yk(t), is

given by

yk(t) =

∫ ∞

−∞rk(α)h(t− α)dα,

=

∫ ∞

−∞rk(α)s

∗(α + τ − t)dα. (3.21)

Substituting (3.20) into (3.21) we have

yk(t) =

∫ ∞

−∞σts(α− τRT )e

j2πkϑtej2πfDαs∗(α + τ − t)dα. (3.22)

Substituting α− τRT by β in (3.22) we have

yk(t) = σtej2πkϑtej2πfDτRT

∫ ∞

−∞s(β)ej2πfDβs∗(β + τRT + τ − t)dβ,

(3.23)

which can be written in terms of the waveform ambiguity function [12],

χ(t, f) =

∫ ∞

−∞s(β)s∗(β − t)ej2πfβdβ, (3.24)

as

yk(t) = σtej2πkϑtej2πfDτRTχ(t− τRT − τ, fD).

(3.25)

The output corresponding to the m-th pulse of the burst, yk,m(t), is done

DBD



by considering the round-trip time as τRT +mTPRI in (3.25), as

yk,m(t) = σtej2πkϑtej2πfD(τRT+mTPRI)χ(t− τRT − τ −mTPRI , fD),

= σ′tej2πkϑtej2πfDmTPRIχ(t− τRT − τ −mTPRI , fD), (3.26)

where σ′t = σte

j2πfDτRT . Assuming that the pulse time-bandwidth product

and the expected target Doppler frequencies are such that the waveform is

insensitive to target Doppler shift, the following approximation is valid

χ(t, fD) ∼= χ(t, 0) =

∫ ∞

−∞s(β)s∗(β − t)dβ. (3.27)

Thus, yk,m(t) becomes

yk,m(t) = σ′tej2πkϑtej2πfDmTPRIχ(t− τRT − τ −mTPRI , 0). (3.28)

Since χ(t, 0) has a maximum for t = 0, then the maximum of |yk,m(t)| occursfor t = τRT + τ +mTPRI and is given by

max |yk,m(t)| = |σt||χ(0, 0)| = |σt|∫ ∞

−∞|s(β)|2dβ, (3.29)

that due to the pulse waveform normalization,∫ ∞

−∞|s(β)|2dβ = 1, (3.30)

is simplified to |σt|.Sampling yk,m(t) at instants t = t0 + lTf + mTPRI , where t0 is the

beginning of the range gate of interest, Tf is the sample interval that determines

the range resolution cell and l, l = 0, . . . , L−1, is known as the fast time index,

yields the samples

yk,m[l] = σ′tej2πkϑtej2πfDmTPRIχ(t0 + lTf − τRT − τ, 0), (3.31)

or equivalently,

yk,m[l] = αlej2πkϑtej2πfDmTPRI , (3.32)

k = 0, . . . , K − 1,

m = 0, . . . , J − 1,

l = 0, . . . , L− 1,

where αl, which depends on l, is given by

αl = σ′tχ(t0 + lTf − τRT − τ, 0). (3.33)

DBD



The magnitude of the samples in (3.32) is given by

|yk,m[l]| = |σt||χ(t0 + lTf − τRT − τ, 0)|, (3.34)

which is maximal for the l∗ which satisfies

t0 + l∗Tf ∼= τRT + τ, (3.35)

where τRT corresponds to the target range, so, as expected, the maximum

happens exactly at the target range. In the case where the equality in (3.35) can

be achieved and considering the pulse waveform normalization, the maximum

is given bymax |yk,m[l]| = |yk,m[l∗]| = |αl∗| = |σt|. (3.36)

For a given l, which corresponds to the l-th range gate, the values of

yk,m[l] can be organized in a K × J matrix. The L× J ×K structure formed

by L range samples of each of the J pulses of all K elements is called the radar

datacube and may be visualized in Fig. 3.4. Each datacube corresponds to a

single CPI. Though the radar doesn’t store data like this, the datacube is a

good representation in order to help the understanding of the STAP process.

The processor is able to beamform along the column dimension and Doppler

process along the rows.

Examination of (3.32) shows that one exponential term depends on the

spatial index k and the other depends on the temporal index m. We can write

a target space-time steering vector, sst, as in [1], where, for a given range gate,

αlsst = [y0,0, . . . , yK−1,0, y0,1, . . . , yK−1,1, . . . , y0,J−1, . . . , yK−1,J−1]T , and

sst = st(fD)⊗ ss(ϑt), (3.37)

where ⊗ is the Kronecker product operator, ss(ϑt) ∈ CK×1 is the spatial array

manifold vector given by

ss(ϑt) = [1, ej2πϑt, ..., ej2π(K−1)ϑt ]T , (3.38)

and st(fD) ∈ CJ×1 is the temporal steering vector, defined by

st(fD) = [1, ej2πfDTPRI , ..., ej2π(J−1)fDTPRI ]T . (3.39)

The radar detection is a binary hypothesis problem, where the null

hypothesis, H0, corresponds to target absence and H1 corresponds to target

presence. The STAP algorithms are applied to every range of interest, which

corresponds to a slice J × K of the CPI data cube. Each range slice can be

stacked to form the M × 1, M = KJ , space-time snapshot, r[l], [1, 13], and

after detection filtering each range slice is tested under both hypothesis.

Under hypothesis H1 of target presence, each space-time snapshot r at

DBD



l-th range gate

L rangegates

Antennaelements, K

PRI, JPulses

fast time axisslow time axis

Figure 3.4: Radar CPI datacube.

the l-th range, r[l] ∈ CM×1, is then given by the sum of the desired signal αlsst,

interference, i[l], clutter, c[l] and noise, n[l],

r[l] = αlsst + i[l] + c[l] + n[l]︸︷︷︸

x[l]

. (3.40)

The target space-time steering vector sst is described in (3.37). The target’s

return power isE[|αl|2] = σ2nξl, where σ

2n is the thermal noise power per element

and ξl is the signal to noise ratio (SNR).

The vector n[l] ∈ CM×1 in (3.40) is the complex vector of sensor noise,

which is assumed to be a zero-mean spatially and temporally circular complex

Gaussian process. The space-time covariance matrix of the noise is given by

Rn = E[n[l]nH [l]

]= σ2

nIM , (3.41)

where IM is the M ×M identity matrix.

The jamming interference, i[l] ∈ CM×1, considered here consists of bar-

rage noise jamming originated from land-based or airborne platforms far from

the radar impinging on the ULA. We assume that the radar pulse repetition

frequency (PRF), fr = 1/TPRI , is significantly less than the instantaneous jam-

mer’s bandwidth so that the jamming is temporally uncorrelated from pulse

to pulse and we assume that the jammer’s bandwidth is small compared to the

carrier frequency so the jammer’s signal is spatially correlated from element

to element.

The simple jammer model described here is based in [11] and neglects

jammer correlation in fast-time. Assuming there are Q jammers in the envir-

onment, the interference component at the l-th range, i[l], in (3.40), is given

by the sum of the contribution of the Q jammers as

DBD



i[l] =

Q∑

q=1

iq[l]. (3.42)

The contribution of the q-th jammer at the l-th range, iq[l], is given by

iq[l] = aq ⊗ ss(ϑq), (3.43)

where aq = [aq,1, . . . , aq,J ] is a random vector containing the q-th jamming

signal’s amplitudes for all J pulses and ss(ϑq) is the q-th jammer spatial

steering vector. Assuming that the jammer samples from different pulses are

uncorrelated and that the jamming signal is stationary over a CPI, thus

E[aqa

Hq

]= σ2

nξqIJ , (3.44)

where ξq is the q-th received jammer to noise ratio (JNR) at an element and

σ2n is the single channel noise power.

The total jammer space-time covariance matrix, Rj, is then given by

Rj = IJ ⊗Φj, (3.45)

where Φj is the total jammer spatial covariance matrix given by

Φj = A(ϑ)ΞjAH(ϑ), (3.46)

where A(ϑ) = [ss(ϑ1), ..., ss(ϑQ)] ∈ CK×Q is the complex matrix composed of

the interferers spatial steering vectors at spatial frequencies ϑ1, ..., ϑQ and Ξj

is the diagonal matrix with the jammer’s power given by

Ξj = diag[σ2nξ1, σ

2nξ2, . . . , σ

2nξQ]. (3.47)

The clutter component, c[l], is originated by the echoes from the environ-

ment, the earth’s surface, known as ground clutter. The clutter is distributed

in both angle and range and is spread in Doppler frequency due to the platform

motion. The clutter component consists of the superposition of returns from

all ambiguous ranges within the radar horizon. In the model assumed here we

suppose unambiguous range [1].

As an approximation to a continuous field of clutter, the clutter return

from a single range will be modeled according to [1] as the superposition

of a large number, Nc, of independent clutter sources that are uniformly

distributed in azimuth about the radar. The location of the (l, p)-th clutter

patch is described by its azimuth, φp, and range, Rl, (or elevation, θl). The

corresponding spatial frequency is

ϑl,p =d

λccos θl sin φp. (3.48)

DBD



The Doppler frequency of the l, p-th patch will be denoted fl,p. The clutter

component of the space-time snapshot at the l-th range is then given by

c[l] =

Nc∑

p=1

αl,pv(ϑl,p, fl,p), (3.49)

where αl,p is the random amplitude from the l, p-th clutter patch and

v(ϑl,p, fl,p) is the space-time steering vector of the l, p-th clutter patch given

byv(ϑl,p, fl,p) = bl,p(fl,p)⊗ al,p(ϑl,p), (3.50)

where al,p(ϑl,p) is the corresponding steering vector of the l, p-th clutter patch

with spatial frequency ϑl,p and bl,p(fl,p) is the corresponding temporal steering

vector of the l, p-th clutter patch with Doppler frequency fl,p.

The contribution from the l, p-th clutter patch has a clutter to noise

ratio (CNR) given by ξl,p. The clutter amplitudes satisfy E [|αl,p|2] = σ2nξl,p.

Assuming that the returns from different clutter patches are uncorrelated and

assuming unambiguous range, only range Rl is contributing to the clutter, the

clutter space-time covariance matrix is given by

Rc = E[c[l]cH [l]

]= σ2

n

Nc∑

p=1

ξl,pv(ϑl,p, fl,p)vH(ϑl,p, fl,p). (3.51)

An aircraft platform motion induces a very special structure to the clutter

due to the dependence of the Doppler frequency on angle. Assuming that the

velocity vector is aligned with the array axis, the clutter Doppler frequency is

given byfl,p =

2vaλc

cos θl sinφp, (3.52)

where va is the platform velocity, which in terms of normalized Doppler,

ωl,p = fl,pTPRI , and spatial frequency is given by

ωl,p = fl,pTPRI =

(2vaTPRI

d

)

ϑl,p. (3.53)

From (3.53), we can note that normalized Doppler is linear in spatial frequency,

more specifically the azimuth sine, sinφp. With normalized coordinates the

slope of the clutter line isβc =

2vaTPRId

, (3.54)

which represents the number of half-interelement spacings traversed by the

platform during one PRI. For half-wavelength interelement spacing, βc =

4va/λcfr is equivalently the number of times the clutter Doppler spectrum

aliases into the unambiguous Doppler space. Equation (3.53) defines the

relation of the presence of clutter in the angle-Doppler space. This is referred

to as the clutter ridge.

DBD



ϑl,p

-0.5 0 0.5

ωl,p

-0.5

0

0.5v

a = 0 m/s, β = 0

ϑl,p

-0.5 0 0.5

ωl,p

-0.5

0

0.5v

a = 25 m/s, β = 0.5

ϑl,p

-0.5 0 0.5

ωl,p

-0.5

0

0.5v

a = 50 m/s, β = 1

ϑl,p

-0.5 0 0.5

ωl,p

-0.5

0

0.5v

a = 125 m/s, β = 2.5

Figure 3.5: Example of clutter ridges for βc = 0, βc = 0.5, βc = 1.0 and βc = 2.5for a PRF = 300 Hz.

A clutter ridge of βc = 1 corresponds to the clutter exactly filling the

Doppler space once. In general, the clutter ridge may span a portion of the

Doppler space, or the whole Doppler space, depending on the platform velocity,

the operating wavelength, and the radar PRF. Fig. 3.5 depicts the default

case of a stationary platform (βc = 0), an unambiguous clutter (βc ≤ 1),

the clutter filling the Doppler space once (βc = 1) and a Doppler ambiguous

clutter (βc > 1). In the Doppler ambiguous case the clutter spectrum extends

over a region larger than the PRF and folds over (aliases) into the observable

Doppler space. In this case there may be multiple angles at which sidelobe

clutter has the same Doppler as a target. Furthermore, as βc increases, the

mainlobe clutter occupies a larger portion of the Doppler space. The more

Doppler-ambiguous the clutter, the more difficult it will be to suppress it.

The total noise-clutter-interference covariance matrix R = E[x[l]xH [l]

]

is given byR = Rc +Rj +Rn, (3.55)

where Rc is given in (3.51), Rj is given in (3.45) and Rn is given in (3.41).

The output of l-th range slice of the datacube, stacked into the space-time

vector r[l] after filtering by a space-time filter w is given by

z[l] = wHr[l], (3.56)

where w = [w1, ..., wM ]T ∈ CM×1 is the complex weight vector of the space-

time filter.

The optimal space-time filter weight vector, w, that maximizes the prob-

ability of detection for a given PFA [14] in a gaussian interference environment

takes the formw = βR−1sst, (3.57)

DBD



where R is the autocorrelation matrix of the clutter, jammers and noise as in

(3.55), and β is a complex constant. In [14], it is proven that maximizing the

SIR also maximizes the probability of detection. Therefore, different choices

of β do not affect the probability of detection, since they don’t alter the SIR.

Nevertheless, wisely choosing β leads to some advantages that will become

clear in the next section. If we set

β =1

sHstR−1sst

, (3.58)

then the space-time filter is given by

w =R−1sst

sHstR−1sst

, (3.59)

which is the solution to the space-time minimum variance distortionless

response (MVDR) design criterion. The optimal space-time filter normalized

as in equation (3.59) is called the MVDR space-time filter.

3.3 Space-Time Metrics

In this section we will explain the most important figures of merit and

metrics used to evaluate the performance of space-time filters.

(a) Adapted Pattern

Given the space-time filter w, its response as a function of angle and

Doppler is one indicator of the processor performance. That is called the

adapted pattern and is given by

P (ϑ, f) = |wHs(ϑ, f)|2, (3.60)

where s(ϑ, f) is given by

s(ϑ, f) = st(f)⊗ ss(ϑ), (3.61)

where ss(ϑ) is the spatial steering vector of the form of (3.38) and st(f) is the

temporal steering vector of the form of (3.39).

The adapted pattern is a two-dimensional angle-Doppler response. The

analogous of the adapted pattern in beamforming is the beampattern. Ideally,

the adapted pattern has nulls in the directions of interference sources and high

gain at the angle and Doppler of the presumed target direction. The shape and

sidelobe levels of the adapted pattern are also of interest. The adapted pattern

generated by using the matched space-time steering vector, w = sst, where

sst = st(ft)⊗ ss(ϑt), (3.62)

DBD



where ft and ϑt are the Doppler and spatial frequency of the target respectively,

is commonly called the quiescent pattern. The quiescent pattern is the

analogous to a simple Capon beamforming [15] at the target spatial angle

followed by pulse integration. This process is optimal in the absence of clutter

and jammers. Note that the quiescent space-time filter given by w = sst is

not data dependent in the sense that it does not need data from the returned

echoes for its generation. Given an arbitrary Doppler and spatial frequency it

is possible to construct the quiescent filter. Its construction doesn’t take into

account neither the clutter nor the jammers of the scenario.

Now we will show a few simulation results to give the reader a feeling

about what we discussed so far. We simulated an airborne radar with K = 18

elements displaced in a ULA separated by half of the carrier wavelength, which

transmits J = 18 pulses per CPI with a PRF of 300 Hz. We included two

broadband jammers (broadband in Doppler spectrum) located at ϑ1 = −0.64

and ϑ2 = 0.42 with jammer to noise ratio (JNR) per element of 38 dB and

clutter uniformly distributed in azimuth, composed of Nc = 360 patches with

clutter to noise ratio (CNR) per element per pulse of 47 dB. A target was

introduced at angle ϑt = 0 and Doppler frequency ft = 100 Hz. Fig. 3.6

shows the Capon spectrum of the described interference scenario (the target

is absent). The Capon spectrum, SC(ϑ, f), is the inverse of the maximum SIR

achievable, as a function of target spatial and Doppler frequencies,

SC(ϑ, f) =1

sH(ϑ, f)R−1s(ϑ, f)). (3.63)

The clutter is in the diagonal and the jammers are the two vertical lines.

Fig. 3.7 shows the normalized quiescent pattern using the known target’s

space-time steering vector w = sst in (3.60) and Fig. 3.8 shows the principal

cuts at the target’s coordinates in angle and Doppler. Fig. 3.9 shows the nor-

malized adapted pattern of the space-time MVDR filter of (3.59) obtained with

perfect knowledge of the interference (clutter, noise and jammer) covariance

matrix in (3.60) and Fig. 3.10 shows the principal cuts at the target’s coordin-

ates in angle and Doppler. Analyzing Figs. 3.7, 3.8, 3.9 and 3.10 we can’t really

foresee the output SIR. It is possible to guess, though, by the position of the

blue lines in Fig. 3.9, that the MVDR space-time filter severely attenuates the

jammers and clutter.

(b) SIR and SIR Loss

A common metric used for analysing a space-time filter performance

is the output signal-to-interference-ratio (SIR), where by interference it is

understood the contribution of clutter, jammers and noise. Given the received

DBD



Figure 3.6: Capon spectrum of R.

Figure 3.7: Normalized quiescent pattern.

sin(Azimuth)-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Rel

ativ

e P

ower

(dB

)

-80

-60

-40

-20

0Doppler Frequency = 100 Hz

Doppler Frequency (Hz)-150 -100 -50 0 50 100 150

Rel

ativ

e P

ower

(dB

)

-80

-60

-40

-20

0Azimuth = 0 deg

Figure 3.8: Principal cuts of the normalized quiescent pattern.

DBD



Figure 3.9: normalized adapted pattern of the space-time MVDR with perfectknowledge of the space-time covariance matrix.

sin(Azimuth)-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Rel

ativ

e P

ower

(dB

)

-80

-60

-40

-20

0Doppler Frequency = 100 Hz

Doppler Frequency (Hz)-150 -100 -50 0 50 100 150

Rel

ativ

e P

ower

(dB

)

-80

-60

-40

-20

0Azimuth = 0 deg

Figure 3.10: Principal cuts of the normalized adapted pattern given by thespace-time MVDR with perfect knowledge of the space-time covariance matrixat the target coordinates.

signal as (3.40) under the target presence hypothesis, the output of filter w is

given byz[l] = wHsstαl +wHi[l] +wHc[l] +wHn[l]

︸︷︷︸

wHx[l]

. (3.64)

The SIR for a target towards direction ϑt and Doppler frequency ft is defined

asSIR =

E[|wHsstαl|2]E[|wHx[l]|2] =

σ2nξl|wHsst|2wHRw

. (3.65)

Substituting the optimum weight vector (3.57) into (3.65) leads to the optimum

SIR given bySIRop = σ2

nξlsHstR

−1sst. (3.66)

For the described example, considering ξl = 1, the quiescent space-time

filter led to a SIR of -17.3 dB, while the MVDR space-time filter led to a SIR of

DBD



25 dB, which is almost the maximum gain achievable for a matched space-time

filter in a noise-only scenario, that is 10 log(ξlKJ) = 10 log(18 · 18) = 25.1

dB. The dramatic difference between the results of both filters reveals the

importance of data dependent space-time filters, and also corroborate the fact

that just by visualizing the adapted pattern it is quite difficult to have an idea

of what the SIR output will be.

Another common metric used in space-time literature is the SIR loss,

namely LSIR, defined asLSIR =

SIR

SIR0, (3.67)

where SIR0 is the maximum SNR achievable in a noise-only environment with

an array of K elements and integration of J pulses, which is [1]

SIR0 = ξlKJ. (3.68)

(c) Doppler Space Performance

The SIR is obtained for a specific angle and Doppler. If the target

velocity is unknown, the interest in SIR performance is a function of the target

Doppler. The metric used to evaluate the Doppler space performance is the SIR

considered by holding the target spatial frequency fixed and varying the target

Doppler, computing a separate space-time filter for each Doppler frequency, f .

In practice, a separate space-time filter is computed for every potential target

Doppler, forming a bank of space-time filters that cover the Doppler space.

The number of filters is typically equal to J , the number of pulses in one CPI.

Let fj be the Doppler which the j-th space-time weighting vector, wj , is tuned

to. Then, the Doppler performance can be computed as

SIR(fj) =σ2nξl|wH

j s(ϑt, fj)|2wHj Rwj

, (3.69)

where s(ϑt, fj) is the space-time steering vector formed as in (3.61), wj is equal

to s(ϑt, fj) for the quiescent space-time filter and equal to the one defined in

(3.59) substituting sst for s(ϑt, fj) for the MVDR space-time filter.

The Doppler performance is usually presented using the SIR loss, com-

puted asLSIR(fj) =

SIR(fj)

SIR0, (3.70)

where SIR0 is defined in (3.68).

Fig. 3.11 shows the Doppler performance of the quiescent and the MVDR

space-time filters. We can see that the MVDR achieves about 0 dB SIR loss over

the majority of the Doppler space, which is near the maximum gain on target,

due to its ability to suppress both clutter and jamming, while the quiescent

filter shows a poor performance. When the target is near 0 Hz or any multiple

DBD



Target Doppler Frequency (Hz)-150 -100 -50 0 50 100 150

SIR

Los

s (d

B)

-70

-60

-50

-40

-30

-20

-10

0

MVDR space-time filterQuiescent space-time filter

Figure 3.11: SIR Loss vs. Doppler frequency for quiescent space-time filter andMVDR space-time filter, fixed ϑ = 0.

of 300 Hz, which is the radar PRF, the performance of the MVDR space-time

filter degrades greatly. That is because the target falls into the null the filter

places to attenuate the mainlobe clutter in both angle and Doppler.

(d) Probability of Detection

In this section we will compute the probability of detection, PD, for a

given space-time filter, w, using the following test,

|wHr[l]| ≷H1H0T, (3.71)

where the quantity |wHr[l]| is called test statistic, r[l] is defined in (3.40) and

T is a comparison threshold. If the test statistic is greater than T , the target

is said to be present (hypothesis H1) and if it is smaller than T the target is

said to be absent (null hypothesis, H0).

The threshold T can be found in terms of the probability of false alarm,

PFA. The test is the magnitude of the complex random variable, z = wHr. We

assume as usual that aq in (3.43) is a Gaussian vector and variables αl,p in

(3.49) are Gaussian [1]. Under hypothesis H0, when the target is absent, the

variable x = |z| has the Rayleigh pdf given by [16]

px(X|H0) =

{2XXe−(X2)/X , X ≥ 0

0, X < 0, (3.72)

where X = E {|z|2} = wHRw and R is defined in (3.55). Thus PFA is given

byPFA =

∫ ∞

T

2X

XeX

2/XdX = e−T2/wHRw. (3.73)

From equation (3.73) we have the threshold, T ,

DBD



T =√

−wHRw ln(PFA). (3.74)

Under hypothesis H1, when the target is present, the variable x = |z| has theRician pdf given by [16]

px(X|H1) =

2XXe(X

2+µ2)/XI0

(√2µ2

XX

)

, X ≥ 0

0, X < 0

, (3.75)

where µ = |σn√ξlw

Hsst| and I0(·) is the modified Bessel function of first order.

Thus PD is given by

PD =

∫ ∞

T

2X

Xe(X

2+µ2)/XI0

(√

2µ2

XX

)

dX. (3.76)

After manipulation it is possible to write (3.76) in terms of the Marcum’s Q

function, QM(α, γ), [12]

PD = QM

(√

2µ2

X,

√

2T 2

X

)

, (3.77)

whereQM(α, γ) =

∫ +∞

γ

Xe−(X2+α2)/2I0(αX)dX. (3.78)

Substituting X = wHRw and µ = |σn√ξlw

Hsst| into (3.77), we have

PD = QM

(√

2σ2nξl|wHsst|2wHRw

,

√

2T 2

wHRw

)

. (3.79)

Substituting T given in equation (3.74) into (3.79) and noticing that SIR =σ2nξl|wHsst|2

wHRw, we can rewrite (3.79) as

PD = QM

(√2SIR,

√

−2 ln(PFA))

, (3.80)

which is the probability of detection, PD, for a given probability of false alarm,

PFA, and a given space-time filter, w, that leads to a certain SIR.

(e) Adaptive Matched Filter (AMF)

The radar system tests each range gate for a series of discrete points over

the spatial and Doppler frequencies of interest, generating hypothesized space-

time steering vectors as in equation (3.61), s(ϑn, fm). The processor usually

chooses several guesses of ϑn and fm equally spaced over a range of spatial

and Doppler frequencies respectively. Since the target real pair (ϑt, ft) is likely

to be different from the tested ones, this mismatch causes the phenomenon of

straddle loss.

DBD



A popular test statistic criterion is given by [17]

|wHr[l]|2 ≷H1H0

T, (3.81)

where w is the space-time filter, r[l] is defined in (3.40) and T can be found in

terms of the probability of false alarm, PFA. The test statistic is the magnitude

square of the complex random variable, z = wHr. Under hypothesis H0, when

the target is absent, the variable x = |z|2 has the exponential pdf given by [16]

px(X|H0) =

{1XeX/X , X ≥ 0

0, X < 0, (3.82)

where X = E {|z|2} = wHRw. Thus, the PFA is given by

PFA =

∫ ∞

T

1

XeX/XdX = e−T/w

HRw. (3.83)

From equation (3.83) we have the threshold, T ,

T = −wHRw ln(PFA). (3.84)

Substituting the space-time filter given in equation (3.57) into the formula of

T in equation (3.84), we have

T = −β2sHstR−1sst ln(PFA), (3.85)

If we choose a normalization factor β as

β =1

√

sHstR−1sst

, (3.86)

and makew =

R−1sst√

sHstR−1sst

, (3.87)

it follows that the test statistic becomes

|wHr[l]|2 ≷H1H0T ′, (3.88)

whereT ′ =

−sHstR−1sst ln(PFA)

sHstR−1sst

= − ln(PFA). (3.89)

From equation (3.89) we can notice that the threshold does not depend on

the data, it is a constant that depends only on the desired PFA. The space-

time filter with the normalization of (3.86) and test statistic of the magnitude

square as in (3.81) shows a constant false alarm rate (CFAR) behavior even

for a fixed threshold. The use of a fixed threshold for a desired probability of

false alarm is a desirable feature in practical systems. The normalized filter w

is also known as the space-time adaptive matched filter (AMF).

We can visualize this test statistic as an output power range profile, that

DBD



Range (km)148 150 152 154 156 158 160

Out

put P

ower

(dB

)

-20

-15

-10

-5

0

5

10

15

20

25

30

optimal AMFThreshold

Figure 3.12: Normalized output power range profile.

shows the test statistic for each l-th range. We expect the output power range

profile to show a peak at the range where the target is present, this peak should

be high enough to be discernable when using a threshold for target detection.

Fig. 3.12 shows the output power range profile for the AMF space-time

filter, whose normalized adapted pattern is shown in Fig. 3.9. For this scenario,

a target was introduced at range 154 km and PFA = 10−4. We can see that the

result of the output power range profile shows a clear peak at range 154 Km,

as expected.

DBD


4Reduced Rank Array Processing

Reducing the dimensionality of the received data in array signal pro-

cessing (e.g. beamforming and space-time processing) is becoming increasingly

more essential, mainly due to the increasing number of sensor elements in

antenna arrays, which generate prohibitively large data sets. Another issue

is the small sample support available in practice for estimation of statistical

quantities [1].

Most uses of STAP are related to the problems of detection, tracking or

imaging [12] in radars. In this thesis, we concern ourselves with the detection

problem in airborne radar systems that employ electronically scanned antennas

(ESA).

In order to detect targets, i.e. uncover objects against noise, ground clut-

ter and jamming, it is interesting to work with both spatial and temporal

domains. Space-time adaptive processing (STAP) involves adaptively (or dy-

namically) adjusting the two-dimensional space-time filter response in an at-

tempt to maximize output signal to interference ratio (SIR), where interference

encompasses everything different from the desired signal, and consequently, im-

proving radar detection performance. Succinctly stated, most classical STAP

algorithms consist of the following steps [18]:

1. Estimating interference covariance matrix and target complex amplitude;

2. Forming a weight vector based on the inverse covariance matrix, the

space-time filter;

3. Calculating the inner product of the space-time filter and the data vector

from a cell under test;

4. Comparing the test statistic based on the value obtained in the former

step with a threshold determined according to a specified false alarm

probability.

From a practical standpoint the key issues associated to these steps include

[18]:

– Sufficient target-free training data support to form an estimated inter-

ference covariance matrix;

DBD


Chapter 4. Reduced Rank Array Processing 53

– Non-singular estimated covariance matrix to form an weight vector;

– Computational complexity in forming the weight vector;

– The ability to maintain a constant false alarm rate (CFAR) and robust

detection performance.

Reduced rank techniques are able to mitigate the nocuous effects of insufficient

target-free training data support to form the estimate of the interference

covariance matrix, which leads to a singular matrix that cannot be inverted,

which in its turn prevents the formation of the space-time filter. Reduced rank

techniques may also reduce dramatically the computational complexity, while

achieving better SIR than full rank techniques in some situations [1].

The Minimum Variance Distortionless Response (MVDR) beamforming

filter [10] is a well-known beamforming technique, also applied in STAP, based

on the inverse of the covariance matrix. The MVDR filter exploits the second-

order statistics of the interference vector to minimize the array variance while

constraining the array response towards the direction of the signal of interest

(SOI). The Minimum Power Distortionless Response (MPDR) beamforming

filter, using the notation of [10], exploits the second-order statistics of the

received vector to minimize the array output power while constraining the

array response towards the direction of the SOI. When the direction of arrival

(DOA) of the SOI and the interference statistics are known exactly the output

of the MPDR reduces to the output of the MVDR.

There are many techniques to implement the MVDR filter with less com-

putational load, basically to avoid the MVDR matrix inversion step, e.g. the

the stochastic-gradient (SG) [19, 20] technique. But full rank algorithms usu-

ally require a large number of snapshots to reach steady-state. In large arrays

arrangements, this may cause degradation in convergence speed, especially in

environments where a small support of independent and identically distributed

(IID) samples are available for estimation of the statistical quantities [1].

Reduced-rank techniques can mitigate the effects of these drawbacks.

Principal Components (PC) [21, 22] and Cross-Spectral Metric (CSM) [13] are

examples of reduced-rank filtering schemes based on eigen-decomposition. The

Multistage Wiener Filter (MWF) achieves rank reduction through the Krylov

subspace, which has the added benefit of a further reduction in computational

complexity based for example on the Lanczos [23, 24, 25] or the Arnoldi [26, 25]

algorithms, or a conjugate gradient-based (CG) implementation [27, 28], CG-

MWF. Yet another rank-reducing algorithm is the Auxiliary Vector Filter

(AVF) [29, 30] that generates a sequence of linear auxiliary filters that converge

to the MVDR filter.

DBD



But rank reduction methods that are optimal in a statistical sense, like

PC, CSM, MWF and AVF may suffer degradation from subspace leakage

in small sample support scenarios [31]. Therefore suboptimal strategies to

reduce the dimensionality of the received data, like the family of adaptive joint

iterative optimization (JIO) algorithms [32, 33], are receiving increasingly more

attention.

4.1 Interpolation-and-Decimation-based Dimensionality ReductionScheme

In the following Chapters, 5 and 6, we apply a dimensionality reduction

technique based on a joint interpolation and decimation scheme applied to

beamforming and space-time processing respectively.

This dimensionality reduction strategy is composed by two stages. The

first stage interpolates the array snapshots in order to generate correlation

between its samples using a finite impulse response (FIR) filter. The second

stage is the decimation stage, which eliminates some samples, reducing the

snapshots’ length. A notable point of this strategy is the elegant and effective

way to design the interpolation filter. The design is such that, for a given decim-

ation pattern, the interpolation filter maximizes the signal-to-interference-and-

noise ratio (SINR) after the decimation stage.

Interpolation and decimation algorithms were extensively studied for

sampling rate alteration or related applications [34]. In these algorithms,

the filter is applied prior to the decimation stage in order to avoid aliasing.

As the sampling rate is an important cost factor in digital signal processor

implementation, so is the length of the input data an important cost factor in

ever increasing sophisticated algorithms.

Professor Raimundo Sampaio Neto, in CETUC, started to investigate the

interpolation and decimation concept focusing on the dimensionality reduction

of the observed data prior to the digital processing algorithms. The idea was

to design the interpolation filter aiming at the minimization of a certain

cost function. At first, this idea originated an algorithm where the rank

reducing stage is coupled with the final application filter (the detection filter

for example), adaptively adjusting both the interpolation and detection filter

weights. This idea proved to give excellent results at low computational cost

in DS-CDMA communication scenarios using MMSE filters [35, 36, 37, 38].

Building on this concept, another strategy was later proposed: designing

a filter that maximizes the signal-to-noise ratio (SNR) after the decimation

stage, completely decoupled from the final application. The advantage of this

stand-alone dimensionality reduction block is that it can be applied prior to

DBD



any desired application filter, for example, any kind of detection or estimation

filter, without the need of redesigning it from the cost function, as it has to

be done for different applications in [35, 36, 37, 38]. This work was tested in

UWB communication scenarios, and had excellent results [39, 40, 41].

An evolution of this work was carried on leading to the design of a filter

that maximizes the signal-to-interference-and-noise ratio (SINR), instead of

only the SNR as in [39, 40, 41]. The design of this new interpolation filter

is creative and effective. This method was applied for DS-CDMA and UWB

communication scenarios and reported, in Portuguese, in [42] and [43]. This

method will be now on called: joint interpolation and decimation scheme

(JIDS).

DBD


5JIDS Dimensionality Reduction Applied to Beamforming

In this chapter, we present the JIDS, which is a dimensionality reduc-

tion technique based on a joint interpolation and decimation scheme, and we

specialize it here for beamforming applications. Investigation of the method

considering the particularities of the beamforming signal model led to simpli-

fications of the method, which allowed for a significant reduction of its overall

complexity. Comparisons with renowned robust and rank reducing techniques

show that the proposed approach has an excellent SINR loss performance with

superior robustness and low computational complexity.

In this chapter, we recast the JIDS algorithm for beamforming applic-

ations and specialize it considering the particularities of this new scenario.

We propose simplified methods for selecting the algorithm parameters and for

defining the best decimation strategy: we investigate the dependence of two

of the JIDS parameters, the decimation factor, F , and the interpolation filter

length, Lv, and obtain a straightforward method to set the length of the in-

terpolation filter. We also derive a new low complexity criterion for selecting

the best decimation pattern. The proposed specialized JIDS for beamforming

will be referred to as JIDSB.

Another concern in beamforming is robustness, which indicates how

the algorithms perform under certain unfavorable situations, e.g. calibration

errors, look of direction errors, distortions caused by source spreading and poor

estimation of statistical quantities due to small sample support. This problem

may be especially dramatic when the MPDR beamformer filter is applied.

Many approaches have been proposed to improve beamformers robustness, for

example, diagonal loading, linear point or derivative constraints, [44, 19, 45,

46], robust estimation using random matrices theory [47] and eigenspace-based

robust beamformers [48]. It is within this context that the JIDSB algorithm is

inserted.

This chapter is organized as follows. Section 5.1 describes the signal

model. Section 5.2 combines the rank reduction transforms with the beam-

former detection filter. Section 5.3 recasts the JIDS, specialized here for

the beamforming application. The proposed simplification procedures are ex-

plained in Subsection 5.3.2 and 5.3.3. Subsection 5.3.4 addresses the computa-

DBD


Chapter 5. JIDS Dimensionality Reduction Applied to Beamforming 57

tional complexity issue. The performance assessment examples of the proposed

specialized and simplified JIDS (JIDSB) are provided in Section 5.4. We apply

the Minimum Variance Distortionless Response (MVDR) [10] detection filter

in the subspace created by the dimensionality reduction stage and compare the

JIDSB results in terms of SINR loss with the full rank MVDR filter and other

rank reduction techniques such as Principal Components (PC) [21, 22], Cross-

Spectral Metric (CSM) [13] and the conjugate gradient-based (CG) Multistage

Wiener Filter (MWF) [27, 28]. In terms of robustness, we compare the JIDSB

with the former methods with diagonal loading when the MPDR [10] filter is

applied. Finally, conclusions are given in Section 5.5.

5.1 System Model

We consider a beamforming application with a uniform linear array

(ULA) with M elements. The sensor array vector received at the i-th time

snapshot r[i] ∈ CM×1 is given by

r[i] = s(θ0)b0[i] + i[i] + n[i]︸︷︷︸

x[i]

, (5.1)

where without loss of generality, the signal of interest is represented as b0,

a complex random variable with power E[|b0|2] = σ20 and steering vector

s(θ0) = [1, e−j2πdλc

sin(θ0), ..., e−j2π(M−1)d

λcsin(θ0)]T , where λc is the carrier wavelength

and d is the inter-element spacing of the ULA, as depicted in Fig. 2.3. The

term x[i] ∈ CM×1 is the interference plus noise vector, where

i[i] = A(Θ)b[i], (5.2)

Θ = [θ1, ..., θQ]T ∈ RQ×1 is the vector of directions of arrival of Q (Q < M) nar-

rowband interference signals impinging the ULA, A(Θ) = [s(θ1), ..., s(θQ)] ∈CM×Q is the complex matrix composed of the interference steering vectors,

s(θk) ∈ CM×1, k = 1, ..., Q, given by:

s(θk) = [1, e−j2πdλc

sin(θk), ..., e−j2π(M−1)d

λcsin(θk)]T . (5.3)

The elements of vector b[i] = [b1[i], ..., bQ[i]]T ∈ C

Q×1 are modeled as

random variables from uncorrelated zero-mean, circular complex processes,

with variances given by σ21, σ

22 , ..., σ

2Q. The vector n[i] ∈ CM×1 is the complex

vector of sensor noise, which is assumed to be a zero-mean spatially and

temporally circular complex Gaussian vector. The beamformer output is then

given byz[i] = wHr[i], (5.4)

where w = [w1, ..., wM ]T ∈ CM×1 is the complex weighting vector.

DBD



The beamformer weighting vector w can be designed to maximize the

signal-to-interference-plus-noise ratio (SINR) in z[i], according to the minimum

variance distortionless response criterion (MVDR) [10]

minwwHRw, subject to wHs(θ0) = 1, (5.5)

where R = E[x[i]xH [i]] is the autocorrelation matrix of the interference and

noise. The well-known solution of (5.5) is the optimal weighting vector given

by:

wop =R−1s(θ0)

sH(θ0)R−1s(θ0). (5.6)

It is easy to show that the SINR achieved with the optimal filter of (5.6) is

given bySINR(wop) = σ2

0sH(θ0)R

−1s(θ0). (5.7)

When the autocorrelation matrix is not known a priori it has to be

estimated from the observed data. In statistical stationary signal scenarios,

the autocorrelation matrix can be estimated from the available sample support

with Ns snapshots as

R =1

Ns

Ns∑

i=1

x[i]xH [i]. (5.8)

The solution in (5.6) that entails the computation of the inverse of the

estimated autocorrelation matrix, R, is called the MVDR sample matrix

inversion (SMI) beamformer [10].

The optimization problem that arises when we use the autocorrelation

matrix of the whole incoming signal, R = E[r[i]rH [i]], is known as minimum

power distortionless response (MPDR) [10]. The MPDR and the MVDR

solutions are identical in conditions of perfect knowledge of the autocorrelation

matrix and the desired steering vector. But SMI-based MPDR beamformers are

known to suffer from performance degradation [10, 44, 49]. The performance

degradation is due to signal cancelation, termed as signal self-nulling. This

problem becomes especially dramatic in practical scenarios, when there are

mismatches between the assumed array response and the true array response.

This situation arises, for example, when there is a finite sample support for

estimating the covariance matrix.

Diagonal loading is a popular approach to improve the MPDR-SMI

beamformer robustness, it is derived by imposing an additional quadratic

constraint either on the Euclidian norm of the weight vector itself or on its

difference from a desired weight vector, [44]. The estimated autocorrelation

matrix added with a diagonal loading γ ∈ R+, RDL is given by

RDL = R+ γI. (5.9)

DBD



M × 1 D × 1T

D × 1

wD

r[i] rD [i] = Tr[i] zD = wHD rD [i]

Figure 5.1: Block diagram of rank reduction stage followed by MVDR beam-former filter.

5.2 Rank Reduction Technique for Beamforming Application

Consider a rank reducing transformation matrix T ∈ CD×M . The

observed i-th snapshot rD[i] ∈ CD×1 after the rank reduction, given by

rD[i] = Tr[i], (5.10)

is then processed by the beamforming filter,wD ∈ CD×1, to produce the output

zD[i], given byzD[i] = wH

DrD[i]. (5.11)

The complex weighting vector wD = [w1, ..., wD]T ∈ CD×1 is designed

according to the MVDR criteria for the reduced observation rD and is given

by

wD =R−1D sD(θ0)

sHD(θ0)R−1D sD(θ0)

, (5.12)

where RD = TRTH = E[xD[i]xHD [i]] is the autocorrelation matrix of the

interference plus noise after the rank reducing stage, with R ∈ CM×M the

autocorrelation matrix of the interference plus noise of the original data and

sD(θ0) = Ts(θ0) is the desired signal steering vector after the rank reducing

stage. The block diagram of this process is illustrated in Fig. 5.1.

5.3 JIDS Rank Reduction Technique

The JIDS rank reduction technique was presented in a regional conference

in Portuguese [42] for DS-CDMA and UWB communications. For the sake

of completeness, this section gives an overview of the method adjusting it

for the beamforming system model. In general terms, the JIDS is based on

two operations: interpolation and decimation, as depicted in Fig 5.2. At the

interpolation stage the i-th received snapshot r[i] is filtered by v ∈ CLv×1

(Lv << M) in order to correlate its components before the decimation stage.

The decimation stage is implemented by means of a decimation matrix,D, that

selects certain components, reducing the original dimension M by a factor of

F . The resulting vector length is D = ⌊M/F ⌋, where ⌊x⌋ is the operation that

selects the largest integer not greater than x. For a uniform decimation by a

factor of F , there are in fact F possible patterns, l ∈ {0, ..., F − 1}. The indexl, that designates the decimation pattern, Dl ∈ CD×M , corresponds to the row

of the first component of r[i] selected by the decimation block, that is

DBD



M × 1

v D

D × 1

Interpolation stage

Decimationstage

Rank Reduction Stage

r[i] rD [i]

Figure 5.2: Illustration of the JIDS rank reduction stage.

Dl =

φl,0

φl,1...

φl,D−1

, (5.13)

whereφl,i = [0, · · · , 0

︸︷︷︸

iF+l

, 1, 0, · · · , 0]. (5.14)

The JIDS rank reducing technique makes a joint choice of the interpol-

ation filter, vl, and the decimation pattern, Dl as will be explained in the

following subsection.

(a) An Effective Design of the Interpolation Filter Specialized forBeamforming

Given a decimation pattern, l, we seek the interpolation filter, v∗l , that

maximizes the SINR at the output of the rank reduction stage. Furthermore,

we also seek the decimation pattern, l∗, that results in the highest SINR among

all F possible decimation patterns, l ∈ {0, . . . , F−1}. The problem of choosing

the decimation pattern, l∗, will be shown to simplify to a trivial comparison

of scalars.

The output of the rank reducing stage using the l-th decimation pattern,

at the i-th snapshot, is given by

rDl[i] = DlVlr[i] (5.15)

= DlVl (s(θ0)b0[i] + i[i] + n[i]) (5.16)

= sDlb0[i] + iDl[i] + nDl[i], (5.17)

where Vl ∈ CM×M is a Toeplitz matrix that implements the discrete convolu-

tion between vl ∈ CLv×1 and r[i] ∈ CM×1. The first column of Vl is given by

[vTl , 0, ..., 0]T ∈ CM×1. Due to the convolution commutation property,

Vlr[i] = R[i]vl, (5.18)

DBD



where R ∈ CM×Lv is a Toeplitz matrix whose first column is given by

r[i] ∈ CM×1. Using (5.18) we can rewrite (5.15) as

rDl[i] = DlR[i]vl. (5.19)

Similarly, sDl , iDl [i] and nDl [i] are defined as

sDl = DlVls(θ0) = DlSvl, (5.20)

iDl [i] = DlVli[i] = DlI[i]vl, (5.21)

nDl [i] = DlVln[i] = DlN [i]vl, (5.22)

where S ∈ CM×Lv , I ∈ CM×Lv and N ∈ CM×Lv are Toeplitz matrices with

their first columns given respectively by s(θ0) ∈ CM×1, i[i] ∈ C

M×1 and

n[i] ∈ CM×1.

The SINR after the rank reducing stage using the l-th decimation pattern

is given by (dropping the snapshot index i for convenience)

SINRl =σ20 ‖ sDl ‖2

E[‖ iDl + nDl ‖2]. (5.23)

The filter v∗l that maximizes (5.23) is the one that satisfies

v∗l = argmax

v

‖ sDl ‖2E[‖ iDl + nDl ‖2]

, (5.24)

which is equivalent to

v∗l = argmax

v

‖ sDl ‖2E[‖ rDl ‖2]

, (5.25)

since

E[‖ rDl ‖2]σ20E[‖ sDl ‖2]

=E[‖ sDlb0 + iDl + nDl ‖2]

σ20 ‖ sDl ‖2

=σ20 ‖ sDl ‖2 +E[‖ iDl + nDl ‖2]

σ20 ‖ sDl ‖2

= 1 +1

SINRl

. (5.26)

The numerator in (5.25) may be written as

‖sDl‖2 = ‖DlSvl‖2 (5.27)

= vHl Alvl, (5.28)

DBD



where Al ∈ CLv×Lv is the symmetric, Hermitian, non-negative matrix given by

Al = SHDHl DlS (5.29)

= SHdiag(pl)S. (5.30)

The matrix diag(pl) is a diagonal matrix that is formed with the elements

of the vector pl along the main diagonal. The vector pl identifies the l-th,

l ∈ {0, ..., F − 1}, decimation pattern: it has zeros at the positions where the

elements will be discarded and ones where the elements will be selected, that

is, the elements with indices {l, l + F, l + 2F, ..., l + ⌊M/F ⌋} are retained.

The denominator in (5.25) may be written as

E[‖ rDl ‖2] = E[‖ DlRvl ‖2] (5.31)

= vHl Blvl, (5.32)

where Bl ∈ CLv×Lv is a hermitian symmetric, non-negative matrix given by

Bl = E[RHDHl DlR] (5.33)

= E[RHdiag(pl)R]. (5.34)

The maximization problem in (5.25) can be restated as

v∗l = argmax

vfl(v), (5.35)

where fl(v) is defined as

fl(v) =vHAlv

vHBlv= vHAlv(v

HBlv)−1. (5.36)

The gradient of fl(v) is computed as

∇vfl(v) = −vHAlv(vHBlv)

−2Blv + (vHBlv)−1Alv. (5.37)

The values that null (5.37), or equivalently,

−(vHAlv)Blv + (vHBlv)Alv = 0, (5.38)

must satisfy

Alv = λBlv, (5.39)

or

Flv = λv, (5.40)

DBD



where Fl = B−1l Al and λ is the scalar given by

λ =vHAlv

vHBlv= fl(v). (5.41)

We can notice that (5.40) is the eigenvalue equation of matrix Fl. Therefore,

vector v that solves (5.40) must be the eigenvector of Fl associated to λ. But

as can be seen comparing (5.41) with (5.36), λ is the SINR itself! Thus, in

order to maximize the SINR we need to find the eigenvector associated to the

largest eigenvalue of Fl. By doing so, we are choosing the interpolation filter,

v∗l , that will produce the maximal SINR for the l-th decimation pattern!

That is the solution to our problem of seeking the interpolation filter

v∗l that maximizes the SINR after the decimation stage. Now, what about

our problem of seeking the decimation pattern, l∗, that results in the highest

SINR among all F decimation patterns? The answer is simple, we just have to

compare the largest eigenvalue of Fl for l ∈ {0, . . . , F − 1} and select l∗ as the

decimation pattern that produces Fl∗ with the largest eigenvalue. This process

can be done in a parallel multiple branch structure with F branches, where

each branch uses a different decimation pattern followed by a simple scalar

comparison. By doing so, we are choosing the interpolation filter v∗l∗ , that will

produce the maximal SINR among all F decimation patterns!

We can now recast the JIDS for beamforming applications:

1. Construct the Toeplitz matrix of the desired steering vector s(θ0), S;

2. Compute the Lv × Lv matrix Al = SHdiag(pl)S, as in (5.30);

3. Estimate the Lv × Lv matrix Bl in (5.34) using NB snapshots as

Bl =1

NB

NB∑

i=1

RH [i]diag(pl)R[i]; (5.42)

4. Compute the Lv × Lv matrix Fl = B−1l Al;

5. Compute the largest eigenvalue, λmax,l, of Fl;

6. Repeat steps 2 to 5 for the F possible decimation patterns and choose

the decimation pattern l∗ that provides the largest eigenvalue of Fl

l∗ = argmaxlλmax,l, (5.43)

l ∈ [0, .., F − 1]. (5.44)

DBD



7. Choose the interpolation filter, v∗l∗ , that corresponds to the decimation

pattern, l∗, which is the eigenvector associated to the largest eigenvalue

of Fl∗ .

After determining D and V using the steps described above, the JIDS

rank reducing transformation matrix TJ ∈ CD×M is, thus, given by

TJ = DV. (5.45)

(b) Selection of the Interpolator Length

The step of selecting the interpolator length, Lv, and decimation factor,

F , may require an extensive search. In this section we will examine the

particularities of the beamforming scenario in order to suggest a good selection

for those parameters adjustments.

Previous researches in scenarios where the observed data is corrupted

only by white noise [39], showed that the best results occurred for interpolation

filter lengths equal to the decimation factor, Lv = F . This may be explained

by the fact that, Lv = F is the filter length that combines the largest number

of samples while preserving the statistical characteristics of the white noise

vector. For this choice of Lv, the time interval between the preserved noise

samples is greater than the memory of the interpolator filter and the white

noise vector remains white after filtering and decimation operations.

That is a good starting point for investigating the parameters setting in

beamforming scenarios as well. It is to be expected that in cases where the

jammer-to-noise-ratio (JNR) is very low, using Lv = F is the best choice, as it

approaches the white noise scenario. In scenarios where the JNR is very high it

may not be the best setting, but it may still be a good choice. We checked this

through extensive computer simulations and verified that, for beamforming

applications, setting Lv = F is indeed a good setting. In this subsection we

will show only two representative results for illustration purposes.

We simulated a beamforming scenario consisting of: M = 64 elements;

signal of interest (SOI) at 0o and SNR = 10 dB. We varied the number of

jammers and their JNR and evaluated the SINR loss for the decimation factors

of 2, 4, 8 and 16 for a range of filter lengths. The SINR loss (LSINR) was

computed as

LSINR =‖wH

DsD‖2(wH

DRDwD)(sH(θ0)R−1s(θ0)), (5.46)

where R is the true autocorrelation of the noise and interference known a

DBD



Interpolator length0 5 10 15 20 25 30 35 40 45

SIN

R L

oss

(dB

)

-2.5

-2

-1.5

-1

-0.5

0

0.5

jammers at -30o, 50o and 65o with JNR = -9 dB

SINR Loss, optimalJIDS with F=16JIDS with F=8JIDS with F=4JIDS with F=2

Lv=2, F=2 Lv=4, F=4

Lv=8, F=8Lv=16, F=16

Figure 5.3: SINR loss as a function of the filter lengths, Lv, for differentdecimation factor, F , for an array with M = 64 elements, SNR = 10 dBand 3 jammers with JNR = -9 dB.

priori and

RD = TJRTHJ , (5.47)

wD =R−1D sD(θ)

sHD(θ)R−1D sD(θ)

, (5.48)

sD = TJs(θ0), (5.49)

where TJ is defined in (5.45). The number of training samples for estimating

Bl is NB = 128, averaged over 200 Montecarlo trials.

In Fig. 5.3 we simulated three jammers impinging on the ULA at angles

−30o, 50o and 65o. All jammers have a JNR of -9 dB. One can see that

for all decimation factors, F , the lowest SINR losses occur for filter lengths

set as Lv = F , as we expected. The arrows in the figure indicate where the

interpolator lengths are equal to the decimation factors. The presentation of

the SINR loss in dB is equivalent to 10 log10 (LSINR).

In Fig. 5.4 we simulated four jammers impinging on the ULA at angles

−60o, −30o, 10o and 50o. All jammers have a high JNR of 15 dB. One can

see that for all decimation factors, except for F = 4 (which is not the best

reduction factor for this scenario anyway), the lowest SINR losses occur for

filter lengths Lv = F . The arrows in the figure indicate where the interpolator

lengths are equal to the decimation factors.

This procedure of setting LV = F avoids the additional step of jointly

optimizing the filter length and the decimation factor. We, therefore, only need

to optimize the decimation factor.

DBD



Interpolator length5 10 15 20 25 30

SIN

R L

oss

(dB

)

-6

-5

-4

-3

-2

-1

0

1

jammers at -60o, -30o , 10o and 50o with JNR = 15 dB

SINR Loss, optimalJIDS with F=16JIDS with F=8JIDS with F=4JIDS with F=2

Lv=2, F=2

Lv=4,

F=4

Lv=8, F=8 Lv=16, F=16

Figure 5.4: SINR loss as a function of the filter lengths, Lv, for differentdecimation factor, F , for an array with M = 64 elements, SNR = 10 dBand 4 jammers with JNR = 15 dB.

(c) JIDS Simplification for Beamforming Environment - JIDSB

Deeper investigation of the ULA structure into the JIDS revealed that

the JIDS can be further simplified. In this context, we propose a low complexity

criterium for selecting the best decimation pattern for ULAs. Considering the

structure of the steering vector in a ULA, we can, instead of selecting the

decimation pattern related to the largest eigenvalue, λmax,l, of Fl, among all

decimation patterns, l ∈ {0, .., F − 1}, select the decimation pattern, l∗, that

corresponds to the largest trace of Fl, denoted tr(Fl),

l∗ = argmaxl

{tr(Fl)} , (5.50)

l ∈ {0, . . . , F − 1}. (5.51)

This procedure leads to similar performance and avoids the need of eigenvalue

decompositions during the decimation pattern decision process. The trace of

Fl (which is equal to the sum of all eigenvalues of Fl) is approximately equal to

the largest eigenvalue of Fl, because, when using the length of the interpolation

filter Lv equal to the reduction factor F , the rank of Fl is at most two, meaning

that Fl has at most two non-zero eigenvalues.

Proof : The rank of Fl = B−1l Al is upper bounded by the minimum between

the rank of B−1l and Al. Since Bl is invertible, Bl is a full rank matrix, leaving

us with the analysis of Al. Matrix Al can be written as

Al = AHDlADl, (5.52)

DBD



where

ADl = DlS. (5.53)

The application of the JIDS in beamforming allows us to use the structure of

the steering vector s(θ0) = [s0, s1, . . . , sM−1]T to go deeper into the structure

of ADl. Matrix S ∈ CM×Lv is Toeplitz i.e.,

S =

s0 0 . . . 0

s1 s0 . . . 0...

......

...

sLv−1 sLv−2 . . . s0...

......

...

sM−1 sM−2 . . . sM−Lv

, (5.54)

with its element, Sp,q, at the p-th row, p ∈ {0, . . . ,M − 1}, and q-th column,

q ∈ {0, . . . , Lv − 1}, formed by

Sp,q ={

sp−q, p ≥ q

0, otherwise.(5.55)

The m-th element, sm, of the steering vector s(θ0) ∈ CM×1 corresponds to the

signal impinging on the m-th antenna element and is given by

sm = e−j 2πdλc

m sin(θ0), m ∈ {0, . . . ,M − 1}. (5.56)

Substituting (5.56) into (5.55),

Sp,q =

{

e−j 2πdλc

(p−q) sin(θ0), p ≥ q

0, otherwise,(5.57)

or equivalently

Sp,q =

{

eαpe−αq, p ≥ q

0, otherwise,(5.58)

where

α = −j2πd

λcsin(θ0). (5.59)

DBD



We can then rewrite the “tall” matrix S as

S =

eα0e−α0

0 . . . 0

eα1e−α0

eα1e−α1

. . . 0

.

.

....

.

.

....

eα(Lv−1)

e−α0

eα(Lv−1)

e−α1

. . . eα(Lv−1)

e−α(Lv−1)

.

.

....

.

.

....

eα(M−1)

e−α0

eα(M−1)

e−α1

. . . eα(M−1)

e−α(Lv−1)

. (5.60)

Matrix S ∈ CM×Lv has at most Lv linearly independent rows. Indeed, from

(5.60), we note that theM−Lv last rows of S are linearly dependent: they can

be expressed as the multiplication of the row vector, [e−α0, e−α1, . . . , e−α(Lv−1)],

by a complex scalar. This means that the Lv linearly independent rows of Sare precisely the first Lv rows of S.

After decimation using the l-th uniform pattern, the i-th row of the

resulting matrix ADl, denoted ADl(i, :), corresponds to the (iF + l)-th row of

matrix S, denoted S(iF + l, :),

ADl(i, :) = S(iF + l, :), (5.61)

i ∈ {0, . . . , D − 1}l ∈ {0, . . . , F − 1}.

Using Lv = F , the l-th uniform decimation pattern, l ∈ [0, . . . , F − 2] has two

linearly independent rows, while decimation pattern l = F − 1 has only one

linearly independent row. Therefore, the rank of Fl is limited by the rank of

Al which in turn is limited by the rank of ADl, consequently the rank is at

most two. This finishes the proof that Fl has at most two nonzero eigenvalues.

As a result, we can choose the Fl, l ∈ {0, · · · , F−1}, which has the largest

trace (which is equal to the sum of all eigenvalues), instead of the one that has

the largest eigenvalue without any meaningful performance degradation. �

To compute the trace of Fl, we can use the fact that the trace of a matrix

C = AB, with A,B,C ∈ CN×N , is given by tr(C) =∑N−1

i=0 A(i, :)B(:, i),

where A(i, :) denotes the i-th row of A and B(:, i) denotes the i-th column of

B. Thus the trace of Fl is computed as

tr(Fl) =

Lv−1∑

i=0

B−1l (i, :)Al(:, i). (5.62)

A representative result is depicted in Fig. 5.5, showing that the proposed

decimation pattern selection procedure is in good agreement with the original

one. In Fig. 5.5 we simulate a ULA wiht M = 64 elements; signal of interest

DBD



Factor2 4 6 8 10 12 14 16

SIN

R L

oss

(dB

)

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Jammers at -80o, -65o, -40o, -25o, 30o, 45o, 60o and 75o with JNR = 15 dB

SINR Loss, optimal JIDS, L

v = F

JIDSB, Lv = F

Figure 5.5: SINR loss as a function of the decimation factor, F , for the proposedand the optimal decimation pattern selection strategies for an array withM = 64 elements, SNR = 10 dB and 8 jammers with JNR = 15 dB.

(SOI) at 0o, SNR = 10 dB; 8 jammers impinging at angles −80o, −65o, −40o,

−25o, 30o, 45o, 60o and 75o with a JNR of 15 dB each. We selected the

interpolator length, Lv = F , as described in subsection 5.3.2.

The algorithm JIDS specialized for beamforming with the proposed

simplification is named the JIDSB rank reduction algorithm.

(d) Computational Complexity

In this subsection, we discriminate the computational complexity of the

algorithms: JIDS and the JIDSB, which is the JIDS with the specialization

and simplification for beamforming environment described in Subsection 5.3.3.

We also compare the total computational complexity of the JIDSB algorithm

applied for reducing the rank of the MVDR-SMI beamformer (JIDSB-SMI)

against the gradient-based multistage Wiener filter (CG-MWF), which is

known to have a reduced computational complexity.

The main steps of the proposed algorithms take place in a lower di-

mensional subspace, because the practical effect of the decimation matrix Dl

is to select just D lines of S and R[i]. Thus, the computation of matrices

Al = SHDHl DlS and Bl[i] = RH [i]DH

l DlR[i] are reduced to the multiplication

of two matrices of size D×Lv. Table 5.1 shows the computational complexity

of the main parts of the JIDS algorithm and Table 5.2 shows the computational

complexity of the main parts of the JIDSB algorithm.

Fig. 5.6 shows how the computational complexity of the stage of decima-

tion pattern selection is decreased by the simplification described in Subsection

5.3.3 as a function of the reduction factor F . In order to assess the number

of operations required for finding the eigenvector associated with the largest

DBD



Table 5.1: Complex operations of the JIDS.Algorithm main steps Multiplications AdditionsComputation of Al FDL2

v F (D − 1)L2v

Estimation of Bl NBFDL2v + FL2

v NBF (D − 1)L2v + (NB − 1)FL2

v

Inversion of Bl O(FL3v) O(FL3

v)Computation of Fl FL3

v F (Lv − 1)L2v

Eigenvalue decomposition O(FL3v) O(FL3

v)

Table 5.2: Complex operations of the JIDSB.Algorithm main steps Multiplications AdditionsComputation of Al FDL2

v F (D − 1)L2v

Estimation of Bl NBFDL2v + FL2

v NBF (D − 1)L2v + (NB − 1)FL2

v

Inversion of Bl O(FL3v) O(FL3

v)Computation of tr(Fl) FL2

v F (L2v − 1)

Computation of Fl L3v (Lv − 1)L2

v

Eigenvalue decomposition O(L3v) O(L3

v)

eigenvalue of a N × N matrix we used the power method [25], which takes

Nit iterations and involves NitN2 complex multiplications and NitN(N − 1)

complex additions. For both algorithms (JIDS and JIDSB), we set the filter

length equal to the reduction factor, Lv = F and Nit = 5. We can see that the

proposed simplification significantly reduces the number of complex operations

for the stage of decimation pattern selection.

Tables 5.3 and 5.4 show step by step how the JIDSB-SMI and the CG-

MWF algorithms implement the MVDR beamformer respectively. The rank of

the CG-MWF algorithm,DMWF, is the number of basis vectors used to describe

the Krylov subspace of the estimated covariance matrix, R [27]. Tables 5.5 and

5.6 shows the number of operations needed to complete all the steps described

in Tables 5.3 and 5.4. In order to assess the number of operations required to

compute the inverse of a N × N matrix we used the Gauss-Jordan method

that takes 2N3/6 + 3N2/6− 5N/6 complex multiplications and additions.

In Fig. 5.7, according to Tables 5.5 and 5.6, we compare the compu-

tational complexity of the JIDSB algorithm with the CG-MWF for differ-

ent decimation factors, F , for different sizes of sample support, Ns. We set

DMWF = 30, as it leads to good performance in small sample supports, as can

be verified in Section 5.4. The number of elements in the array is set toM = 64,

Nit = 5 and the number of snapshots, NB, used to estimate Bl ∈ CLv×Lv is

the total number of snapshots available, NB = Ns. We can note from Fig. 5.7

that the JIDSB has a remarkably smaller computational complexity than the

CG-MWF for factors F = 2 and F = 4. For F = 8, the complexity is notably

lower for smaller sample supports and it is almost the same as the computa-

DBD



F0 20 40 60 80 100 120 140 160 180

10lo

g 10(N

umbe

r of

Com

plex

Mul

tiplic

atio

ns)

10

20

30

40

50

60

70

80

90

100

JIDSJIDSB

Figure 5.6: Comparison of the number of complex multiplications of the stageof decimation pattern selection of the JIDS and JIDSB for Lv = F and Nit = 5.

Table 5.3: JIDSB-SMI Strategy

initialization RD(0) = 0D×D, FS = Toeplitz(s(θ0))JIDSB for rank reduction

for l = 1, . . . , Fpl = l-th fixed decimation pattern,Al = SHdiag(pl)S,Bl =

1NB

∑NBi=1RH [i]diag(pl)R[i],

where R[i] = Toeplitz(x[i])end

l∗ = argmaxl∑Lv

i=1B−1l (i, :)Al(:, 1),

Fl∗ = B−1l∗ Al∗

v = eigenvector associated with the largest eigenvalue of Fl∗V = Toeplitz(v)D is constructed according to (5.13)MVDR-SMI after rank reduction stage

TD = DVsD = TDs(θ0)

RD = 1Ns

∑Nsi=1TDr[i]r

H [i]THD = TDRTH

D

wD = R−1D sD/s

HDR

−1D sD

DBD



Table 5.4: CG-MWF Strategyinitialization u1 = −g0 = s(θ0), γ0 = ‖g0‖2, w0 = 0M×1, K = D

R = 1Ns

∑Nsi=1 x[i]x

H [i]

CG-MWF

for each iteration k = 1, . . . , K

vk = Rukηk =

γk−1

uHk vk

wk = wk−1 + ηkukgk = gk−1 + ηkvkγk = ‖gk‖2uk+1[i] = −gk[i] +

γk−1

γkuk

Output:

wCG-MWF = wKsH (θ0)wK

Table 5.5: Complex products of the JIDSB-SMI and CG-MWF algorithms.Algorithm Multiplications

(Ns + 1)DM +D2(Ns + 2) + 2D+2D3/6 + 3D2/6− 5D/6

JIDSB-SMI +FDL2v(1 +NB) + L2

v(2F +Nit) + L3v

+2FL3v/6 + 3FL2

v/6− 5FLv/6CG-MWF (Ns + 1)M2 +DMWF(M

2 + 5M + 2) + 4M

Table 5.6: Complex additions of the JIDSB-SMI and CG-MWF algorithms.Algorithm Additions

NsD(M − 1) +NsD2 +D(M − 1)− 1

+2D3/6 + 3D2/6− 5D/6JIDSB-SMI +FDL2

v(1 +NB)− L2v(F + 1)− F +NitLv(Lv − 1)

+L3v + 2FL3

v/6 + 3FL2v/6− 5FLv/6

CG-MWF M2(Ns − 1) +DMWF(M2 + 4M − 2) + 2(M − 1)

tional complexity of the CG-MWF for larger sample supports. For F = 16, the

JIDSB has a significantly greater computational complexity, that is because for

this pair of array size, M , and factor, F , the JIDSB pre-processing steps take

place in a not so reduced subspace, increasing significantly the computational

complexity.

5.4 Simulations

In this section, we compare the performance results in terms of SINR loss

vs. sample support of the JIDSB with renowned rank reducing algorithms. We

apply the MVDR detection filter with and without diagonal loading. In order

to assess the JIDSB robustness, we compare results of the MPDR detection

filter (which is the MVDR solution when the desired signal is present during

estimation of the autocorrelation matrix) for the JIDSB and the other rank

DBD



Sample support50 100 150 200 250 300

10lo

g 10(N

umbe

r of

com

plex

mul

tiplic

atio

ns)

46

48

50

52

54

56

58

60

62

64

66

CG-MWF, DMWF

= 30

JIDSB, F = 2, D = M/F =32JIDSB, F = 4, D = M/F = 16JIDSB, F = 8, D = M/F = 8JIDSB, F = 16, D = M/F = 4

Figure 5.7: Comparison of the number of complex multiplications of theJIDSB with the CG-MWF as a function of the sample support, Ns, forF = {2, 4, 8, 16}, M = 64, Nit = 5, DMWF = 30 and NB = Ns.

reducing algorithms with diagonal loading. We illustrate all the algorithm’s

beampattern and we also simulate a communication application and present

results in terms of bit error rate (BER) performance.

At first, we evaluate the SINR loss performance of the proposed JIDSB-

SMI algorithm and compare it with the performance of the JIDS-SMI, PC-SMI,

CSM-SMI and a gradient-based multistage wiener filter (CG-MWF) algorithm.

The PC-SMI and CSM-SMI algorithms follow the same structure of the JIDS-

SMI as explained in Section 5.2: the received array snapshots go through

the rank reduction stage and then are used to estimate the lower dimension

covariance matrix which is then inverted and used in the computation of the

MVDR or MPDR filter. The rank of the PC-SMI algorithm is the number of

vectors used to form the subspace, which is spanned by the D eigenvectors of

the estimated covariance matrix, R, associated with the D largest eigenvalues

[21]. The rank of the CSM-SMI algorithm is the number of vectors used to

form the subspace, which is spanned by the D eigenvectors of the estimated

covariance matrix, R, that maximizes the cross spectral metric [13]. The rank

of the CG-MWF algorithm is the number of basis vectors used to describe

the Krylov subspace of the estimated covariance matrix, R [27]. The CG-

MWF algorithm converges to the MVDR or MPDR result without the need of

inverting the estimated covariance matrix, R. The rank of the JIDSB-SMI is

⌊M/F ⌋ = D, which is the final length of the vectors after the rank reduction

stage.

The mean SINR loss, LSINR = SINR/SNR0, where the SINR is computed

asSINR =

σ20E{|wHTs(θ0)|2

}

E {wHTRTHw} , (5.63)

DBD



where w is the beamforming filter, T is the rank reducing transformation

(applied when necessary), SNR0 = sH(θ0)R−1s(θ0), R is the true autocorrela-

tion matrix of noise and interference and the expectation , E{·}, is estimated

by averaging 200 Montecarlo runs. Note that although R is the true autocor-

relation matrix, w and T are computed using the estimated R, R, and are,

therefore, stochastic quantities.

In these simulations we adopt a uniform linear array (ULA) consisting

of M = 64 sensor elements whose inter-element spacing is half a signal

wavelength. The signal of interest (SOI) is at 0o. We simulate two scenarios

and compare the algorithms SINR loss for different sample supports. In the

first scenario, there are two narrowband jammers impinging at angles 30o and

50o and for the second scenario, there are 6 narrowband jammers impinging

at angles −65o, −40o, −25o, 30o, 45o and 60o . Each jammer has a JNR of 15

dB.

For both scenarios, we examine the case when the amount of diagonal

loading is set to γ = σ2n in (5.9), just in order to avoid computational

instabilities in the algorithms and the case when the diagonal loading is set

as γ = 10σ2n, which is empirically shown in [50] to be a suitable value. We

compare the former case using two different detection filters: the MVDR, when

the desired signal is not present during estimation of the covariance matrix

and the MPDR, when the desired signal is present during the autocorrelation

matrix estimation.

For all simulations the JIDS and JIDSB had equivalent performances, as

expected, so in the following we will mention only the JIDSB. Figs 5.8 and

5.11 show the SINR loss vs. sample support with a diagonal loading of σ2n. We

can see that the JIDSB had an impressive superior performance compared to

full rank MVDR-SMI, CG-MWF, CSM-SMI and PC-SMI algorithms.

Figs. 5.9 and 5.12 show the SINR loss vs. sample support with a diagonal

loading of 10σ2n. As expected, all algorithms improved their convergence rate

when compared to the case of a diagonal loading of σ2n depicted in Figs 5.8 and

5.11. Even with the great improvement of the full rank MVDR-SMI and CG-

MWF, the proposed JIDSB still had similar performance, with a significantly

lower computational complexity for the case of Fig. 5.12 (L=F=2) and a lower

computational complexity for small sample supports and similar computational

complexity as the CG-MWF for large sample supports for the case of Fig. 5.9

(L=F=8), as can be verified in Fig. 5.7.

Now we examine what happens when the desired signal is present in

the observed data during estimation of the autocorrelation matrix. Figs 5.10

and 5.13 show how the performance is affected by the self-nulling when the

DBD



Sample support50 100 150 200 250 300

SIN

R L

oss

(dB

)

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Jammers at 30o and 50o with JNR = 15 dB and γ = σn2

SINR Loss, OPTIMALFull Rank MVDR-SMICG-MWF rank 30CSM rank = 34PC rank = 50JIDS, L=F=8, rank = 8JIDSB, L=F=8, rank = 8

Figure 5.8: Output SINR loss versus the sample support with M = 64, 2jammers with JNR = 15 dB and γ = σ2

n.

Sample support50 100 150 200 250 300

SIN

R L

oss

(dB

)

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Jammers at 30o and 50o with JNR = 15 dB and γ = 10σn2

SINR Loss, OPTIMAL Full rank MVDR SMICG-MWF rank 30CSM rank = 34PC rank = 50JIDS, L=F=8, rank = 8JIDSB, L=F=8, rank = 8

Figure 5.9: Output SINR loss versus the sample support with M = 64, 2jammers with JNR = 15 dB and γ = 10σ2

n.

MPDR detection filter is applied. As expected, for a high SNR such as 10 dB,

there is a great degradation in performance of all algorithms. The JIDSB had

better performance than all other algorithms. Emphasizing that in Fig. 5.10

the JIDSB showed an impressive superior behavior.

Fig. 5.14 shows the beampattern of all algorithms with the configuration

that led to the results in Fig. 5.12 with a sample support of 200 snapshots. The

arrows indicate the positions of the jammers. All curves were averaged over 200

Montecarlo runs. We can see that all algorithms were able to set deep nulls at

the jammers positions. The JIDSB beampattern follows the same beampattern

as the CG-MWF and the full rank MVDR-SMI.

Fig. 5.15 illustrates the application of the sensor array for a communic-

ation system and show the bit error rate (BER), for all algorithms for a small

DBD



50 100 150 200 250 300

SIN

R L

oss

(dB

)

-25

-20

-15

-10

-5

Jammers at 30o and 50o with JNR = 15 dB and γ = 10σn2

Full rank MPDR SMICG-MWF rank 30CSM rank = 34PC rank = 50JIDS, L=F=8, rank = 8JIDSB, L=F=8, rank = 8

Sample support


n when the desired signal is presentduring estimation of the autocorrelation matrix.

sample support. The system model is as described in (5.1) and the scenario is

again the same related to Fig. 5.12. The simulated desired signal, b0, represents

a symbol of a QPSK constellation with power σ20 = E[|b0|2] and impinges on

the array from direction θ0 = 0o. The outputs of the beamformer filters, z[i] as

in (5.4) and zD[i] as in (5.11), are fed to a minimum-distance QPSK detector,

which is the optimal detector for gaussian channels.

Since noise and jammers are modelled as independent complex Gaussian

random vectors, it is possible to compute semi-analytically the bit error

probability, P(e), asP(e) = E [P (e|SINR(w))] , (5.64)

which can be estimated by the average

BER =1

Np

Np∑

i=1

P (e|SINR(wi)) , (5.65)

whereP(e|SINR(wi)) = Q

(√

SINR(wi))

, (5.66)

withQ (α) =

∫ ∞

α

1√2πe

−t2

2 dt, (5.67)

andSINR(wi) =

σ20 ||wH

i Tis(θ0)||2wHi TiRTH

i wi

, (5.68)

where wi and Ti are the detection filter and the rank reducing transformation

obtained in the i-th simulation run and R is the autocorrelation matrix of the

noise and interference. Fig. 5.15 depict the semi-analytical bit error rate, BER,

of the described QPSK system using Np = 1000 for a sample support of 20. In

the horizontal axis we show the SNR based on the reference detection SNR of

DBD



Sample support50 100 150 200 250 300

SIN

R L

oss

(dB

)

-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Jammers at -65o, -40o, -25o, 30o, 45o and 60o, JNR = 15 dB and γ=σn2

SINR Loss, OPTIMALFull Rank MVDR-SMICG-MWF rank 30CSM rank = 34PC rank = 50JIDS, L=F=2, rank = 32JIDSB, L=F=2, rank = 32

Figure 5.11: Output SINR loss versus the sample support with M = 64, 6jammers with JNR = 15 dB and γ = σ2

n.

Sample support50 100 150 200 250 300

SIN

R L

oss

(dB

)

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

Jammers at -65o, -40o, -25o, 30o, 45o and 60o, JNR = 15 dB and γ=10σn2

SINR Loss, OPTIMAL Full rank MVDR SMICG-MWF rank 30CSM rank = 34PC rank = 50JIDS, L=F=2, rank = 32JIDSB, L=F=2, rank = 32


n.

the optimal full rank case without interference, given as

SNR = 10 log10

(σ20

σ2n

||s(θ0)||2)

= 10 log10

(Mσ2

0

σ2n

)

. (5.69)

We can see that the JIDSB performance is slightly better than the full rank

MVDR-SMI and CG-MWF, that is because the mean SINR for that scenario,

with a sample support of only 20 snapshots is slightly better for the JIDSB

than for the CG-MWF.

DBD



Sample support50 100 150 200 250 300

SIN

R L

oss

(dB

)

-30

-25

-20

-15

-10

Jammers at -65o, -40o, -25o, 30o, 45o and 60o, JNR = 15 dB and γ=10σn2

Full rank MPDR SMICG-MWF rank 30CSM rank = 34PC rank = 50JIDS, L=F=2, rank = 32JIDSB, L=F=2, rank = 32


n when the desired signal is presentduring estimation of the autocorrelation matrix.

θ

-80 -60 -40 -20 0 20 40 60 80

Nor

mal

ized

out

put p

ower

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

Full rank SMICG-MWF, rank = 30CSM, rank = 30PC, rank = 50JIDSB, L=F=2, rank =32

Figure 5.14: Adapted beampattern for different algorithms for sample supportNs = 100, M = 64 and 6 jammers with JNR = 15 dB.

SNR9 10 11 12 13 14 15 16 17 18 19

BE

R

10-6

10-4

10-2

Sample Support = 20

JIDSB L=F=2, rank = 32PC rank = 50CSM rank = 30CG-MWF rank = 30Full Rank MVDR-SMI

Figure 5.15: Semi-analytical BER for different algorithms for sample supportNs = 20, M = 64, 6 jammers with JNR = 15 dB.

DBD



5.5 Conclusion

In this chapter we reported the JIDS rank reducing technique and ad-

apted it for beamforming. We also proposed the JIDSB algorithm, which is a

specialized version of the JIDS technique for beamforming with new proposed

simplification procedures that resulted from analysis of the combination of the

beamforming system model with the JIDS structure. The proposed simplific-

ations reduce its number of operations and complexity without degrading its

performance. The JIDS algorithm consists in a stage of dimensionality reduc-

tion decoupled from the detection filter, based on a joint interpolation and

decimation scheme. The JIDS strategy design is an elegant and effective way

to obtain the joint interpolation filter and decimation pattern, taking advant-

age of the correlation generated by the interpolation filter in order to eliminate

samples and still achieve a high SINR after the decimation stage.

We presented performance results in terms of SINR loss vs. sample

support applying the MVDR filter with and without diagonal loading and

the MPDR filter with diagonal loading. We could see the superiority of the

JIDSB-SMI in terms of SINR loss performance for the MPDR filter and for

the MVDR filter without diagonal loading. In the case of the MVDR filter

with diagonal loading the JIDSB had similar performance as the full rank

MVDR-SMI and CG-MWF, with an impressive low computational complexity

especially for small sample supports.

In our understanding, the proposed JIDSB algorithm shows great advant-

ages in beamforming scenarios. It is an inherently robust method, it has similar

or superior SINR loss performance than the full rank MVDR-SMI and CG-

MWF, low computational complexity and it reduces significantly the length of

the processed data.

DBD


6JIDS Applied to Space-Time Processing

In this chapter we specialize the JIDS rank reduction technique for

space-time applications, more specifically, airborne phased-array radars. We

start by recasting the JIDS algorithm for STAP applications and we use the

method described in Chapter 5 for setting the decimation factor, F , and the

interpolation filter length, Lv.

We compare results of the JIDS algorithm in terms of probability of

detection and Doppler performance with the full rank Minimum Variance

Distortionless Response sample matrix inversion (MVDR-SMI) space-time

filter and other rank reduction techniques such as Principal Components (PC),

Cross-Spectral Metric (CSM) and Multistage Wiener Filter (MWF).

Simulation results show that the proposed approach has an impressive

ability to reduce significantly the length of the space-time snapshots and

to achieve very good performance in Doppler SINR loss and probability of

detection, especially in sample starving scenarios.

The study on this topic carried out during the preparation of this thesis

produced the conference paper [6].

This chapter is organized as follows: the system model is described in

Section 6.1; the proposed technique is explained in Section 6.2; computer

simulations are presented in Section 6.3 and, finally, conclusions are given

in Section 6.4.

6.1 System Model

The system model considered here is explained in detail in Section 3.2.

The output of the i-th range slice of the datacube, stacked into the space-

time vector r[i] defined in (3.40), after filtering by a space-time filter,w, is given

byz[i] = wHr[i]. (6.1)

where w = [w1, ..., wM ]T ∈ CM×1 is the complex weight vector of the space-

time filter. The optimal space-time filter weight vector, w, that maximizes the

probability of detection, PD, for a given PFA takes the form (3.57)

w = βR−1sst,

DBD


Chapter 6. JIDS Applied to Space-Time Processing 81

Figure 6.1: Datacube showing selection of reference and guard cells for estim-ating the covariance matrix for a given cell under test (CUT).

where β is a complex constant, R is the total noise-clutter-interference covari-

ance matrix defined in (3.55) and sst is the target space-time steering vector

defined in (3.37). If we set β = (sHstR−1sst)

−1, then the space-time filter is given

by

w =R−1sst

sHstR−1sst

, (6.2)

which is the solution to the space-time minimum variance distortionless

response (MVDR) design criterion, the MVDR space-time filter. In practice,

the assumption of perfect knowledge of the covariance matrix, R, is not

realistic. It must be estimated from an available finite sample support, Ke,

with Ke elements, asR =

1

Ke

∑

i∈Ker[i]rH [i]. (6.3)

Typically, the training data, r[i], cover a range interval surrounding but not

including the range gate of interest as shown in Fig. 6.1 in order to the target-

free assumption to hold in practice.

The optimum space-time filter in (6.2) replacing R−1 by the inverted

estimated covariance matrix, R−1, is known as the MVDR sample matrix

inversion (SMI) space-time filter. An additional performance loss dependent on

the number of samples is incurred due to the covariance matrix estimation. If

all training data are independent and identically distributed (iid) with respect

to the null hypothesis, choosing Ke ≈ 2JK, where J is the number of pulses

and K is the number of sensors, yields an average performance loss of roughtly

3 dB [51].

The sample support availability becomes more critical because of the

nonstationary nature of real radar clutter and jamming. Clutter heterogeneity

in range, combined with power and elevation angle dependence, reduce the

number of range gates over which the clutter scenario is effectively stationary.

Reduced rank techniques are able to mitigate the deleterious effects of

DBD



small sample support in the space-time covariance matrix estimation. The

reduced rank solution, wD, of (6.2) is given by

wD =R−1D sD

sHDR−1D sD

, (6.4)

where RD = TRTH = E[xD[i]xHD [i]] is the autocorrelation matrix of the

interference plus noise plus clutter after the rank reducing stage, xD[i] = Tx[i],

sD = Tsst is the desired signal space-time steering vector after the rank

reducing stage and finally, T, is the rank reducing transformation.

6.2 Recast of the JIDS applied to STAP

The JIDS rank reduction technique is here specialized for space-time

applications. The JIDS rank reducing technique makes a joint choice of the

interpolation filter, vl, and the decimation pattern, Dl. Given a decimation

pattern, the interpolation filter, v∗l , is the one that maximizes the SINR after

the rank reduction stage.

Now we will briefly recast the process of generating the JIDS interpolator

filter given the decimation pattern, Dl, explained in Section 5.3.1 for space-

time applications:

1. Construct the Toeplitz matrix of the desired space-time steering vector

sst, S;

2. Compute Al = SHdiag(pl)S, as in (5.30);

3. Estimate Bl in (5.34) using NB space-time snapshots as

Bl =1

NB

NB∑

i=1

XH [i]diag(pl)X [i]; (6.5)

4. Compute Fl = B−1l Al;

5. Compute the largest eigenvalue, λmax,l, of Fl;

6. Repeat steps 2 to 5 for the F possible decimation patterns and choose

the decimation pattern l∗ that provides the largest eigenvalue of Fl

l∗ = argmaxlλmax,l, l = {0, .., F − 1}. (6.6)

7. Set the interpolation filter as vl∗ , which is the eigenvector associated to

the largest eigenvalue of Fl∗ . The interpolation filter obtained using this

procedure is the one that maximizes the SINR.

DBD



Figure 6.2: Capon spectrum of R.

The method used here for setting the interpolator length and the decim-

ation factor is explained in Section 5.3.2.

After determining D and V using the steps described above, the JIDS

rank reducing transformation matrix, TJ ∈ CD×M , is given by

TJ = DV, (6.7)

therefore, wD in (6.4) is computed by substituting T by TJ in (6.4).

6.3 Simulations

In this section we show some performance results using the full rank

MVDR-SMI and reduced rank algorithms for sufficient and for small sample

supports. The reduced rank algorithms selected for these examples are the same

as used in Chapter 5: Principal Components (PC) [21, 52], Cross Spectral

Metric (CSM) [53, 54, 13] and the Multistage Wiener Filter (MWF) [27]

algorithms.

We simulated an airborne radar with K = 18 elements displaced in a

ULA separated by half of the carrier wavelength, which transmits J = 18 pulses

per CPI with a pulse repetition frequency (PRF) of 300 Hz. We included two

broadband jammers, nj = 2, located at ϑ1 = −0.64 and ϑ2 = 0.42 with jammer

to noise ratio (JNR) per element of 38 dB and clutter uniformly distributed

in azimuth, unambiguous in Doppler (βc = 1), with no velocity misalignment

and no inner clutter motion [1], composed of Nc = 360 patches with clutter to

noise ratio (CNR) per element per pulse of 47 dB.

Fig. 6.2 shows the Capon spectrum of the described scenario. A target

was introduced at spatial frequency ϑt = 0 and Doppler frequency ft = 100

Hz.

DBD



Eigenvalue Number0 50 71 100 150 200 250 300 350

10lo

g 10(e

igen

valu

e)

-10

0

10

20

30

40

50

60

70Eigenspectra

β = 1, va = 50 m/s

Brennan's rule limit

Figure 6.3: Eigenspectrum of the covariance matrix R.

The rank of the interference covariance matrix, ri, is given by

ri = rj + rc, (6.8)

where rj is the rank of the jammers and rc is the rank of the clutter. The rank

of the jammers, rj , is given by [1]

rj = JnJ , (6.9)

where J is the number of pulses per CPI and nJ is the number of active

jammers. The rank of the clutter, rc, is given approximately by the Brennan’s

rule [1]rc ≃ ⌊K + (J − 1)βc⌋, (6.10)

where K is the number sensors in the array, J is the number of pulses per CPI

and βc is the clutter ridge. When βc is an integer, the equality in (6.10) holds.

Using (6.8), (6.9) and (6.10) with the data used for simulation we have

that the total interference covariance matrix rank is 71. From Fig. 6.3 we can

confirm that, from the 324 total eigenvalues of R, the largest 71 encompasses

the most significant part of the spectrum, thus we set the rank for the PC-SMI

and CSM-SMI algorithms equal to 71. Even though the MWF decomposes the

signal using the Krylov subspace instead of the eigenspace, we also set its total

number of stages to 71 for the sake of comparison.

Fig. 6.4 depicts the SINR Loss (LSINR) vs. the decimation factor, F , for

the JIDS algorithm assuming the length of the interpolator filter Lv = F ,

Lv = ⌊F/2⌋ and Lv = ⌊1.5F ⌋. This example empirically confirms that setting

the interpolator length equals to the decimation factor is a good choice. This

plot shows that we have better performance in terms of LSINR for a decimation

factor and filter length of 54, which reduces the length of the space-time

snapshot from 324 to 6. That is a very impressive reduction!

DBD



Factor0 10 20 30 40 50 60 70 80 90

LS

INR

(dB

)

-60

-50

-40

-30

-20

-10

0

Optimal SINR LossL

v = F

Lv = F/2

Lv = 1.5F

Figure 6.4: SINR Loss (LSINR) vs. decimation factor for the JIDS algorithmaccording the described example and with Ke = 648, ξt = 1, ϑt = 0 andft = 100 Hz.

We added an empirical value of diagonal loading (δ = 10σ2n) used in

practice for the estimate of matrix R for all algorithms in order to improve

robustness and to avoid numerical instabilities when computing the inverse in

(6.2) for small sample support. Figs. 6.5 and 6.6 show the SINR Loss vs. target

Doppler of the optimal MVDR, the full rank MVDR-SMI, PC-SMI, CSM-SMI,

MWF and the JIDS-SMI for a sample support of 648 and 50 respectively.

Curves were averaged over 100 runs. The Doppler performance is computed

by holding fixed the target spatial frequency, ϑt, and evaluating for different

target Doppler frequencies, fj, the SINR loss, LSINR = SINR/SNR0, given by

LSINR(fj) =σ2nξt|wH

j THj s(ϑt, fj)|2

wHj T

Hj RTjwjSNR0

, (6.11)

where wj is the space-time filter computed for a target at spatial frequency,

ϑt, and Doppler frequency, fj, Tj is the rank reducing transformation (applied

when necessary) and SNR0 = ξtKJ .

In Figs. 6.5 and 6.6 we can see that the SINR loss is very large when

the target is near 0 Hz, because in this case the target falls into the filters’

notch used to suppress the mainlobe clutter. We can also see that as the

sample support decreases from one figure to the other, so does the Doppler

performance of all but the JIDS algorithm. The JIDS is probably more

robust to small sample supports because it is able to achieve the largest rank

reduction. The JIDS used a reduced vector of size 6, instead of 71 or 324,

therefore, a sample support of 50 is not so dramatically small.

Figs. 6.7 and 6.8 show the probability of detection, PD, vs. the probability

of false alarm, PFA, for the MVDR-SMI and the reduced rank algorithms,

DBD




SIR

Los

s (d

B)

-80

-70

-60

-50

-40

-30

-20

-10

0Doppler performance with sample support of 648

optimal MVDRMVDR-SMIJIDS-SMI, D = 6CSM-SMI, D = 71PC-SMI, D = 71MWF, D = 71

Figure 6.5: LSINR vs. Doppler with sample support of Ke = 648, fixed ϑt = 0.


SIR

Los

s (d

B)

-80

-70

-60

-50

-40

-30

-20

-10

0Doppler performance with sample support of 50

optimal MVDRMVDR-SMIJIDS-SMI, D = 6CSM-SMI, D = 71PC-SMI, D = 71MWF, D = 71

Figure 6.6: LSINR vs. Doppler with sample support of Ke = 50, fixed ϑt = 0.

computed as [12]

PD = QM

(√2SINR,

√

−2 ln(PFA))

, (6.12)

whereQM(·) is the Marcum’s Q function. It was assessed at ϑt = 0 and ft = 100

Hz for a sample support of 50. The mean SINR attained is also displayed. The

curves were averaged over 100 runs. The SNR was set as ξt = 1. From Fig.

6.8 we can see that the only algorithm that does not collapse for such a low

sample support is the JIDS.

Just for the sake of curiosity we show the adapted pattern of the SMI

algorithms for a sample support of 648 and 50. Figs. 6.9, 6.10, 6.11, 6.12,

6.13 show the adapted pattern for a sample support of 648 of the MVDR-

SMI, JIDS-SMI, MWF, CSM-SMI and PC-SMI algorithms respectively. Figs.

6.14, 6.15, 6.16, 6.17, 6.18 show the adapted pattern for a sample support of

50 of the MVDR-SMI, JIDS-SMI, MWF, CSM-SMI and PC-SMI algorithms

DBD



Probability of false alarm10-10 10-8 10-6 10-4 10-2 100

Pro

babi

lity

of d

etec

tion

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Sample support of 648

MVDR-SMI SINR = 12 dBMWF SINR = 13 dBCSM-SMI SINR = 10 dBPC-SMI SINR = -8 dBJIDS SINR = 14 dB

Figure 6.7: PD vs. PFA for Ke = 648, ξt = 1, ϑt = 0 and ft = 100 Hz, all otherparameters are the same as the former examples.

Probability of false alarm10-10 10-8 10-6 10-4 10-2 100

Pro

babi

lity

of d

etec

tion

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Sample support of 50

MVDR-SMI SINR = -7 dBMWF SINR = -7 dBCSM-SMI SINR = -15 dBPC-SMI SINR = -27 dBJIDS SINR = 13 dB

Figure 6.8: PD vs. PFA for Ke = 50, ξt = 1, ϑt = 0 and ft = 100 Hz, all otherparameters are the same as the former examples.

respectively. It is interesting to note that comparing Fig. 6.16 of the MWF

adapted pattern for a sample support of 50 and Fig. 6.15 of the JIDS adapted

pattern for a sample support of 50, it seems that their adapted patterns are

“similar”, but Fig. 6.8 shows the clear superior performance in probability of

detection of the JIDS.

DBD



Figure 6.9: Pattern of the space-time MVDR-SMI with sample support of 648- SINR = 12 dB.

Figure 6.10: Pattern of the space-time JIDS-SMI with sample support of 648- SINR = 14 dB.

Figure 6.11: Pattern of the space-time MWF-SMI with sample support of 648- SINR = 13 dB.

DBD



Figure 6.12: Pattern of the space-time CSM-SMI with sample support of 648- SINR = 10 dB.

Figure 6.13: Pattern of the space-time PC-SMI with sample support of 648 -SINR = -8 dB.

Figure 6.14: Pattern of the space-time MVDR-SMI with sample support of 50- SINR = -7 dB.

DBD



Figure 6.15: Pattern of the space-time JIDS-SMI with sample support of 50 -SINR = 13 dB.

Figure 6.16: Pattern of the space-time MWF-SMI with sample support of 50 -SINR = -7 dB.

Figure 6.17: Pattern of the space-time CSM-SMI with sample support of 50 -SINR = -15 dB.

DBD



Figure 6.18: Pattern of the space-time PC-SMI with sample support of 50 -SINR = -27 dB.

The next example shows the application of the reduced-rank algorithms

in a real data experiment. We use the Mountain-top data set. This data set

can be downloaded from the internet [55]. The area where the samples were

collected is a desert area, with a river, suburban areas, and mountains within

the field of view. There are 14 sensor elements displaced in a ULA horizontally

oriented with respect to the ground with a inter-element spacing of 0.333m.

Although the radar system is installed at a fixed ground site, radar platform

motion is emulated using the Inverse Displaced Phase Center array (IDPCA).

The data are organized in CPIs of 16 pulses. Here, we used the data file

t38pre01v1 CPI6. The transmitted pulse is a linear frequency modulated chirp.

The stored data has been demodulated, equalized across the system bandwidth

in each channel, and calibrated from channel to channel. The data is not pulse

compressed; satisfactory pulse compression may be obtained by constructing a

matched filter based on the 500 kHz LFM transmit pulse. The range samples

are collected in intervals of 1 µs. The radar PRF is 625 Hz.

This data contains ground clutter primarily from a single large scatterer

(a mountain) which is isolated in angle, 245o, with a Doppler of 156 Hz (due

to platform motion). The target is located at the same range (154 Km) and

Doppler of the clutter with a different azimuth, 275o.

Figs. 6.19, 6.20, 6.21, 6.22, 6.23 and 6.24 depict the normalized output

power of the statistical test |wH [l]sst|2 for all cells under test (CUTs), which

implies in a range of 147 to 162 Km. The covariance matrix, R, is estimated

using a symmetrical sliding window with a total of 20 snapshots, respecting a

guard interval of 3 snapshots before the CUT and 3 snapshots after the CUT.

Fig. 6.19 depicts the unadapted weight vector, which corresponds to the

space-time matched filter w[l] = sst. From Fig. 6.19, we can note that the

DBD



Range (km)148 150 152 154 156 158 160 162

Rel

ativ

e O

utpu

t Pow

er (

dB)

-40

-35

-30

-25

-20

-15

-10

-5

0Mountain Top Dataset with Snapshots:20

unadapted w

Figure 6.19: Range profile for the Mountain Top data set using the unadaptedspace-time filter wQ = sst and a sample support of 20 snapshots.

target is not detectable without space-time processing.

Fig. 6.20 depicts the output of the full rank MVDR-SMI algorithm with

a diagonal loading of δ = 107. After extensive tests, this value was chosen

because it leads to better results. For all other algorithms the same diagonal

loading was applied.

We can see, comparing Fig. 6.22 with Figs. 6.20, 6.21, 6.23 and 6.24,

that the results of the JIDS algorithm is impressively better than the others

for such a small sample support (only 20 snapshots to estimate a covariance

matrix of 224×224). While the others are able to detect the target using a

threshold of around 5 dB less than the peak, the JIDS is able to detect the

target with a threshold of more than 8 dB of difference.

DBD



Range (km)148 150 152 154 156 158 160 162

Rel

ativ

e O

utpu

t Pow

er (

dB)

-40

-35

-30

-25

-20

-15

-10

-5


SMI

Figure 6.20: Range profile for the Mountain Top data set using the full rankMVDR-SMI algorithm with diagonal loading and a sample support of 20snapshots.

Range (km)148 150 152 154 156 158 160 162

Rel

ativ

e O

utpu

t Pow

er (

dB)

-40

-35

-30

-25

-20

-15

-10

-5


MWF rank = 17

Figure 6.21: Range profile for the Mountain Top data set using the MWFalgorithm and a sample support of 20 snapshots.

Range (km)148 150 152 154 156 158 160 162

Rel

ativ

e O

utpu

t Pow

er (

dB)

-40

-35

-30

-25

-20

-15

-10-8.35

-5


JIDS L = F = 14

Figure 6.22: Range profile for the Mountain Top data set using the JIDSalgorithm and a sample support of 20 snapshots.

DBD



Range (km)148 150 152 154 156 158 160 162

Rel

ativ

e O

utpu

t Pow

er (

dB)

-40

-35

-30

-25

-20

-15

-10

-4.44


PC rank = 80

Figure 6.23: Range profile for the Mountain Top data set using the PC-SMIalgorithm and a sample support of 20 snapshots.

Range (km)148 150 152 154 156 158 160 162

Rel

ativ

e O

utpu

t Pow

er (

dB)

-25

-20

-15

-10

-5

-1.330

Mountain Top Dataset with Snapshots:20

CSM rank = 17

Figure 6.24: Range profile for the Mountain Top data set using the CSM-SMIalgorithm and a sample support of 20 snapshots.

6.4 Conclusion

In this chapter we reported the JIDS rank reducing technique specialized

for space-time applications. The JIDS algorithm consists in a stage of

dimensionality reduction decoupled from the space-time target detection fil-

ter, based on a joint interpolation and decimation scheme. This strategy design

takes advantage of the correlation generated by the interpolation filter in or-

der to eliminate samples and still achieve low SINR loss. It is able to work in

a very reduced Krylov subspace. Numerical results obtained for a simulated

sample starving airborne radar scenario showed that the JIDS achieves su-

perior Doppler SINR loss performance and increased probability of detection

when compared to the full rank MVDR-SMI, PC-SMI, CSM-SMI and MWF.

Tests with real airborne data show that the JIDS algorithm has an impressive

DBD



superior detection performance for a real heterogeneous clutter scenario.

DBD


7Radar and Communications

The topic of coexistence of radar and communications has acquired

high importance due to the problem of spectrum encroachment and inter-

ference in passive sensors, [56, 57, 58]. Extensive efforts have been direc-

ted towards this matter and many different alternatives have been proposed

to deal with it, for example, sophisticated interference mitigation techniques

[59, 60], spectrum-sharing for simultaneous operation of radar and communica-

tions [61, 62, 63, 64, 65, 66] and dual-functionality platforms, where waveform

and irradiating sensors are jointly managed for both radar and communication

functions. Pulsed radars enable synchronization for embedded communications

and usually operate over considerable ranges with high power [67]. The com-

bination of this characteristics with the high radar pulse repetition frequency

(PRF) makes pulsed radars able to meet the communication service opera-

tional requirements [56].

The alternative of dual-functionality platform approach is divided into

two categories that can be combined: embedding information into radar

emissions via waveform diversity [68, 69, 70] and embedding information

towards a specific receiver direction by changing the amplitude and/or phase of

the sidelobe transmit beampattern towards that direction [71, 2, 3, 5, 4, 7, 8].

The first category of embedding information into a radar emission by changing

the waveform during each radar pulse may affect the radar main function even

within a coherent processing interval (CPI). We are especially interested in the

second category: embedding communication symbols by changing amplitude

and/or phase of the transmit beampattern.

In this context of varying parameters of the transmit beampattern, we

can also differentiate signaling strategies by the number of simultaneous wave-

forms transmitted during each pulse period and by the type of modulation

applied to the transmit beampattern: a) amplitude modulation or sidelobe

level modulation (SLL), which is equivalent to embedding symbols from an

Amplitude Shift Keying (ASK) constellation, b) phase modulation, which is

equivalent to embedding symbols from a Phase Shift Keying (PSK) constella-

tion, or c) a combination of amplitude and phase modulation, which is equival-

ent to embedding symbols from an Amplitude and Phase Shift Keying (APSK)

DBD


Chapter 7. Radar and Communications 97

constellation. But due to the difficulty of having coherency between the com-

munication receivers and the radar, the work so far in this area is restricted

to non-coherent receivers.

The works of [71, 5, 7, 8] describe different methodologies for embed-

ding SLL modulation into the transmit beampattern. They require only one

waveform, but as shown in [5], if more than one orthogonal waveform can be

transmitted through different antenna arrays, one can use a different transmit

beampattern for each waveform, thus increasing the bit rate (multi-waveform

ASK).

The research in phase modulation is up to now restricted to non-

coherent modes. Non-coherent phase modulation embeds information into the

phase difference, so it requires a reference phase. So far this reference phase

is achieved through the transmission of multiple orthogonal waveforms (at

least two) during the same pulse period through different antenna arrays

(multi-waveform Differential PSK-based methods). Many radars have the

ability to transmit orthogonal waveforms simultaneously through different

antenna arrays, MIMO radars are a good example [72], but it requires more

sophisticated technology than the necessary to change only the transmit

beampattern of only one array of sensors. To cope with this issue, we propose

a signaling strategy that is suitable for more simple radars, that have only one

transmit array.

In the work of [3], during each radar pulse, multiple pairs of orthogonal

waveforms are transmitted, where each pair represents one phase symbol. At

the communication receiver, each embedded symbol is detected by estimating

the phase difference between the signals associated with the two waveforms in

each pair. The phase embedded at the beampatterns are designed using phase

rotational invariance [73]. In the work of [4] several waveforms are transmitted

simultaneously and only one waveform is taken as a common reference to

all other waveforms. Each communication symbol is embedded in the phase

difference between the two signal components associated with each waveform

pair. In [4], the phase embedded at the beampatterns are designed by using

phase rotational invariance [73] or by convex optimization.

The option of using radar for communication purposes is also an al-

ternative to deal with the security issue in defense-related applications, where

it is essential to maintain secure communications with low probability of in-

tercept. The secure behavior of the radar-embedded communication system

based in varying the transmit beampattern parameters in specific directions is

intrinsic to its design. The preassigment of the receiver direction may provide

operational bit error rates (BER) only towards this direction. This advantage,

DBD



though, becomes a major disadvantage if the real receiver position doesn’t

match exactly the predefined one. This situation can easily arise if the com-

munication receiver is moving relative to the radar.

In this thesis we address this problem. We have derived the optimal

closed form solution to an optimization formulation that embeds sidelobe

modulation and is robust to small receivers’ direction errors. Our design uses

a constrained optimization problem to generate transmit beampatterns, that

satisfactorily match a given transmit profile (with high fidelity adjustment at

the mainlobe), where each beampattern embeds a different symbol towards the

communication receiver direction and sustains its value over a small angular

region, making it robust against small angular deviations. In this thesis, we

also propose an alternative way of dealing with the mainlobe adjustment

requirement, while still keeping the robustness. The alternative proposed

method is based on eigenvalue, point and derivative constraints, which also

has a closed form solution.

Another important consideration, other than robustness, which was only

addressed by us so far, is adaptation in real-time applications when the

communication receiver is moving relative to the radar. We simplify our two

proposed methods making them suitable for online real-time processing with

very low computational load for updating the beampatterns for following a

moving communication receiver platform.

In Section 7.1 we describe the system model that is used throughout

the present and the following chapters. In Chapter 8, we give an overview

of the main sidelobe modulation methods present in the literature, showing

their pros and cons. In this overview, we focus only on the central ideas

concerning sidelobe modulation and after addressing important topics we make

brief comments, where the reader can find our independent conclusions about

the described methods. The techniques of the methods in the literature were

adapted to conform to the adopted system model and the notation was made

uniform in agreement with this thesis notation.

In Section 9.1, we elaborate our thoughts about robustness and effects

of relative movement between the communication receiver and the radar. This

section links our proposed methods with the methods present in the literature,

as we face this issue directly. In the next sections of Chapter 9 we describe our

proposed techniques for embedding robust amplitude and phase modulation to

the beampattern sidelobe. We also further develop them to real-time iterative

low-complexity versions.

DBD



(a) What is a Dual Function Radar and Communications (DFRC) Radar?

A DFRC radar is a radar which is also used for communications purposes.

The communication function is seen as a secondary task and shouldn’t cause

degradation to the primary radar function. It is within this context that

radar-embedded sidelobe modulation was conceived. Theoretically, any usual

communication’s modulation can be adapted to this new concept, but in

practice there are many restrictions which limit the overall possibilities. In

the following we will explain the sidelobe modulation concept.

(b) What’s the Idea of Sidelobe Modulation?

Sidelobe modulation within the DFRC context consists in varying para-

meters of the transmit beampattern towards a specific direction (the receiver’s

direction) from pulse to pulse according to the communication message to be

transmitted. The variation of the parameters, i.e. amplitude and/or phase fol-

lows the communication symbol’s constellation and is achieved through the

use of different beamforming weighting vectors.

The low-pass complex envelope of the RF transmitted radar pulse, x(t, u),

irradiated towards direction u = cos(θ), as defined in Section 2.4, due to the

contribution of all elements of the sensor array is given by,

x(t, u) = wHs(u)x(t),

and that

B(u) = wHs(u),

is the beampattern, the goal of sidelobe modulation is that towards the

communication receiver direction, uc = cos(θc), the complex value, that will

multiply the pulse x(t), belongs to a symbol constellation C, which k-th symbol

is Ck, k = 1, . . . , K. In other words, we must impose that

Bk(uc) = wHk s(uc) = Ck, (7.1)

where wk ∈ CM×1 , k = 1, . . . , K is the transmit beamforming weighting

vector that generates the k-th transmit power radiation pattern that embeds

Ck towards uc.

The general idea for a non-coherent ASK modulation is ideally depicted

in Figs. 7.1 and 7.2, where two transmit power patterns, P (u) = |B(u)|2,are depicted. Each power pattern has a different sidelobe level towards the

communication receiver and each level corresponds to one of two symbols of a

non-coherent ASK constellation. The dotted line corresponds to the maximum

DBD



sidelobe level allowed in terms of radar operation. Ideally we want to modify

only the sidelobe related to the communication receiver and keep the rest

unchanged.

Figure 7.1: Transmit power pattern generated by the transmit beamformer w0,which embeds symbol “0” towards the receiver direction.

Figure 7.2: Transmit power pattern generated by the transmit beamformer w1,which embeds symbol “1” towards the receiver direction.

7.1 System Model

We consider a pulsed-Doppler phased array radar with M radiating

elements, assumed to have the same radiation pattern, displaced in a uniform

linear array (ULA) which is used for transmitting purpose. In order to perform

pulse integration, during a CPI,MI pulses are transmitted with a PRF of 1/Tr,

where Tr is the pulse repetition interval (PRI) [12].

As detailed in Section 2.4, the low-pass complex envelope of the RF

transmitted radar pulse, x(t, u), irradiated towards direction u = cos(θ), due

to the contribution of all elements of the sensor array is given by

DBD



Figure 7.3: Coordinate system.

x(t, u) = wHs(u)x(t), (7.2)

where w = [w0, . . . , wM−1]T ∈ C

M×1 is the beamforming weighting vector, s(u)

is the array steering vector pointed at u and x(t) is given by

x(t) =√

2Ptejψs(t), (7.3)

where Pt is the signal power, ψ is the initial carrier phase and s(t) is the

radar waveform, which is a low-pass pulse of duration τ , with bandwidth W ,

normalized to unitary energy.

For the radar and communications topic we use the second example model

of array geometry of Section 2.2, depicted here in Fig. 7.3, and we assume

the steering vector defined in (2.28), in which we consider the inter-element

spacing, d, as d = λc/2 without loss of generality.

DBD


8Overview of the Sidelobe Modulation Methods

In this chapter, we explain in detail the main sidelobe modulation

methods present in the literature, focussing only on their central ideas. We

make several simulations of the methods and a deeper analysis than the original

authors themselves. We also present our independent opinion about the pros

and cons of the existing methods.

8.1 Sidelobe Level (SLL) Modulation

In this section we give an overview of the methods present in the

literature to embed non-coherent amplitude modulation to the radar transmit

beampattern. There are mainly two methods: the method of [71] and the

method of [5]. The method proposed in [71] achieves the design of multiple

transmit power patterns with the same mainlobe and different sidelobe levels

(SLLs) by using time modulated arrays (TMA) and the method proposed by

the Villanova Center of Advanced Communications Lab in [5], extended to a

posterior work in [2], which solves a convex optimization problem by means of

interior point techniques.

The optimization criterion involved in the method of [71] is highly

nonlinear and computationally demanding, making it difficult to generate

multiple transmit power distribution patterns with the same mainlobe. The

situation is even worse if you think of multiple receivers and movement

between the receivers and the radar platform. The other method uses a convex

optimization formulation for this problem, which is much simpler to be solved.

The authors of [2] remain very active in the area and claim that their strategy

can cope with multiple receivers and adaptation in a real-time environment.

This overview will explain in detail their strategy.

The main objective of any SLL modulation strategy is to deliver a

communication message to a receiver (or multiple receivers) located within

the sidelobe region of the radar transmit beampattern as a secondary task

without affecting the radar operation. Therefore, one key requirement is to

keep the magnitude of the mainbeam of the radar the same during the entire

processing interval. On the other hand, in order to embed information in the

beamformer, the SLL at the communication direction(s) should be permitted

DBD


Chapter 8. Overview of the Sidelobe Modulation Methods 103

to assume different values over the regions of interest.

These two key requirements can be achieved via appropriate transmit

beamforming designs. The design described in [5] assumes that the mainbeam,

where the radar task takes place, is defined by the spatial sector U =

[umin, umax]. The special case where the radar main beam is focused towards

a single spatial direction corresponds to umin = umax. The sidelobe area is

denoted as U . Assume that the number of communication receivers is L and

the corresponding communication directions are ul, l = 1, . . . , L.

Let wk, k = 0, . . . , K−1, where K is the total number of communication

symbols, be theM×1 transmit beamforming weight vectors. One way to design

the transmit weight vectors is to formulate the following optimization problem,

minwk

maxu

∣∣|Bd(u)| − |wH

k s(u)|∣∣ , u ∈ U (8.1)

subject to |wHk s(u)| ≤ ǫ , u ∈ U , (8.2)

wHk s(ul) = δl,k , l = {1, . . . , L}, (8.3)

whereBd(u) is the desired beampattern within the main radar beam defined by

the radar designers, ǫ is a positive number of user’s choice used for controlling

the overall SLLs, δl,k is a positive number used to determine the SLL associated

with the k-th transmit beam towards the l-th communication direction and

s(u) is the M × 1 steering vector of the transmit array towards u defined in

(2.28).

In (8.1) - (8.3), the objective function fits all K transmit beampatterns

to the beampattern mandated by the radar operation. The set of constraints

in (8.2) is used to upper-bound the transmit power leakage within the sidelobe

areas, which is also mandated by the radar operation. Note that the upper

bound determined by the parameter ǫ is the same for all transmit beams.

The set of linear constraints (8.3) is associated with the secondary function

of the system which is to embed information via different SLLs towards the

communication directions. It is worth-noting that the parameter δl,k which

determines the SLL is different for each one of the K transmit beams. Since ǫ

is the highest sidelobe level as mandated by the main radar operation of the

system, the conditions|δl,k| ≤ ǫ, k = 0, . . . , K − 1 should be satisfied.

The optimization problem (8.1) - (8.3) is difficult to solve due to the

non-convex objective function. Therefore, the authors of [2, 5] reformulate the

problem by slightly modifying the objective function. They substitute |Bd(u)|in (8.1) by Bd(u) and set Bd(u) = ejφ(u), yielding the following optimization

DBD



problem

minwk

maxui

∣∣ejφ(ui) −wH

k s(ui)∣∣ , ui ∈ U , i = 1, . . . , I (8.4)

subject to |wHk s(up)| ≤ ǫ , up ∈ U , p = 1, . . . , P (8.5)

wHk s(ul) = δl,k , l = {1, . . . , L}, (8.6)

where ui, i = 1, . . . , I, and up, p = 1, . . . , P , are discrete grids of angles used

to approximate U and U , respectively, and φ(u) is a phase profile of user’s

choice. The optimization problem (8.4) - (8.6) is convex and can be solved in

a computationally efficient manner [74]. It is worth noting that the transmit

beamforming weight vector obtained by solving (8.4) - (8.6) yields a unity

magnitude within the main radar beam. However, in practice the transmit

weight vector can be scaled up to the desired transmit gain as long as the

total transmit power budget does not exceed the maximum allowed power of

the actual system. Note that scaling up the transmit weight vector results

in magnifying the transmit power distribution at all angles equally, i.e., the

relative SLLs with respect to the mainlobe remains unchanged.

However, the parameter ǫ should be carefully chosen to guarantee a

feasible solution. One way to do this is to solve the following auxiliary problem

minwk,ǫ

ǫ (8.7)

subject to |wHk s(up)| ≤ ǫ , up ∈ U , p = 1, . . . , P (8.8)

wHk s(ul) = δl,k , l = {1, . . . , L}, (8.9)

which is guaranteed to have a feasible solution. Denote the solution to (8.7) -

(8.9) as ǫmin. Then, the range of values of ǫ which guarantees a feasible solution

to the optimization problem (8.4) - (8.6) is given as ǫ ≥ ǫmin.

(a) Brief Discussion about the Existing SLL Modulation Methods

As we have seen so far, we can note that an optimization problem must

be solved by means of interior point technique each time any of the variables

change. It means, if any of the L communication receivers is moving relative to

the radar and ul, l = 1, . . . , L, changes its value, a new optimization problem

must be solved for the new value of ul. The solution to the new problem can

be achieved efficiently offline, but it takes time and resources that may not be

available for real time applications.

It is also interesting to note that, though the authors of the described

method say that the maximum allowable sidelobe level, ǫ, is defined by the

radar operator, there is no guarantee that the solution to their problem

DBD



will lead to sidelobe levels below, ǫ. They can tell what is the minimum

ǫ achievable given the communication receiver parameters, if this value is

acceptable in terms of radar performance, then the communication function

can be embedded to the radar. But it is important to emphasize that the

minimum ǫ changes as the communication parameters change. It means that,

even if the transmit beampatterns passed the sidelobe test for a given number

of communication receivers at specific directions, it does not guarantee that

they will pass the test again for a different set of directions or different number

of communication receivers.

(b) Symbol Error Analysis

The discussion in this subsection follows a different approach than [5].

In [5], the authors, as radar experts, derive the symbol error expressions

using their own methodology and nomenclature. Though the expressions are

correct, it is possible to present the DFRC SLL modulation using the common

digital communication theory. Here in this subsection we insert the digital

communications platform into the DFRC SLL modulation context, which

allows a reader familiar with digital communications to understand it more

clearly. We generate illustrative examples and come up with some conclusions.

The low-pass complex envelope of the sequence of RF transmitted radar

pulses, x(t, u), irradiated towards direction u = cos(θ), due to the contribution

of all elements of the sensor array, is given by

x(t, u) =√

2Etejψ

∞∑

i=0

ci(u)s(t− iTPRI), (8.10)

where ci(u) ∈ {Ck(u) = wHk s(u)}K−1

k=0 , TPRI is the pulse repetition interval,

Et is the isotropically irradiated RF pulse energy, ψ is the initial carrier

phase and s(t) is the radar waveform, which is a low-pass pulse of duration

τ , with bandwidth W , normalized to unitary energy. The average energy of

the i-th transmitted radar pulse is then 2EtE[|ci(u)|2]. Towards the receiver

communication direction, uc, the information symbol is

ci ∈ {Ck = wHk s(uc)}K−1

k=0 , (8.11)

where Ck is a positive amplitude which corresponds to a symbol from an ASK

modulation.

At the communication receiver located at direction uc of the transmit

beampattern, the output, r[i], after demodulation, matched filtering with the

waveform s(t) and sampling at the point of maximum (see Fig. 8.1), is given

DBD



X

1√2

| · |

Compwiththres-holds

zs∗(τ − t)

r[i] cix(t, u) + n(t)

t = iTPRI + τ

Figure 8.1: Illustration of the non-coherent ASK receiver.

by

r[i] =√

Erciejψ + ni, (8.12)

where Er is the energy, which was isotropically irradiated by the transmitter,

at the receiver side, considering the propagation attenuation effect. Assuming

equally likely symbols, the average energy of the signal term in (8.12) is then

Es = ErE[|ck|2] = Er1

K

K−1∑

k=0

|wHk s(uc)|2. (8.13)

The noise, ni, is a complex Gaussian random variable with zero mean and

variance σ2n = N0. The signal to noise ratio, SNR, is given by

SNR =ErE[|ck|2]

σ2n

=ErN0

1

K

K−1∑

k=0

|wHk s(uc)|2. (8.14)

Dividing and multiplying (8.14) by |B(umax)|2, where B(umax) is the beampat-

tern towards the direction of maximum, we have that the SNR can be written

as

SNR =Er|B(umax)|2

N0

1

K

K−1∑

k=0

|wHk s(uc)|2

|B(umax)|2. (8.15)

But Er|B(umax)|2 is the energy at a receiver located towards the direction of

maximum of the transmit beampattern, Emax, and

|wHk s(uc)|2

|B(umax)|2=

|Bk(uc)|2|B(umax)|2

= |Bk(uc)|2. (8.16)

Thus, the SNR may be written as

SNR =Emax

N0

1

K

K−1∑

k=0

|Bk(uc)|2. (8.17)

As we are dealing with a non-coherent ASK modulation, we take the

absolute value of r[i], z = |r[i]|, and compare to the thresholds. A decision is

made for symbol k if z is inside the region Zk, delimited by the thresholds,

Tk−1,k ≤ z < Tk,k+1. Fig. 8.2 depicts an example of the four decision regions

for an illustrative 4-ASK constellation. In this example, we can define the

DBD



Figure 8.2: Illustration of decision regions for a 4-ASK constellation.

thresholds corresponding to the extremities as T−1,0 = 0 and T3,4 = +∞ and

the other thresholds are set in the median between two adjacent symbols for

illustration purpose only.

The probability of wrong detection, P (E), is given by

P (E) = 1− P (R), (8.18)

where P (R) is the probability of right decision, which is given by

P (R) =K−1∑

k=0

PkP (z ∈ Zk | k) , (8.19)

where Pk is the probability of transmitting the k-symbol and P (z ∈ Zk|k) isthe probability of z being inside the decision region for symbol k, given that

symbol k was transmitted. We can write this conditional probability as

P (z ∈ Zk | k) =∫

Zk

pz|k(Z)dZ, (8.20)

where pz|k(Z) is the conditional probability density of z given that symbol k

was transmitted.

When symbol k is transmitted, r is a complex Gaussian variable with a

non-zero mean, mk, given by mk =√ErCke

jψ and variance given by σ2n, where

the real and imaginary parts of r are statistically independent of each other

with variance given by σ2n = σ2

n/2 = N0/2. Therefore, the conditional density

function of z = |r[i]| given that symbol k is transmitted follows the Rician

probability distribution,

pz|k(Z) =Ze−(Z2+µ2k)/(2σ

2n)

σ2n

I0

(Zµkσ2n

)

, (8.21)

where µk = |mk| andI0(η) ,

1

π

∫ π

0

eη cos θdθ, (8.22)

is the modified Bessel function of the first kind and zero-th order. Substituting

DBD



(8.21) into (8.20) and composing with (8.19) and (8.18), considering that the

K symbols are equally probable of being transmitted, i.e. Pk = 1/K, we have

the analytical error probability for the SLL modulation

P (E) = 1− 1

K

K−1∑

k=0

∫

Zk

Ze−(Z2+µ2k)/(2σ2n)

σ2n

I0

(Zµkσ2n

)

dZ, (8.23)

where µk =∣∣√ErCke

jψ∣∣. Using the Marcum Q function, Q(a, b), defined as

Q(a, b) =

∫ ∞

b

I0(ax)e− 1

2(a2+x2)xdx, (8.24)

and substituting a = µk/σn and x = Z/σn we can rewrite (8.23) as

P (E) = 1− 1

K

K−1∑

k=0

[

Q

(µkσn,Tk−1,k

σn

)

− Q

(µkσn,Tk,k+1

σn

)]

, (8.25)

where Tk−1,k and Tk,k+1 are the thresholds that bound the region Zk, Zk =

[Tk−1,k, Tk,k+1).

It is not so easy to define the thresholds that minimize the error

probability. One approach is to set the thresholds in the point of intersection

between the conditional probability density functions for each symbol, or the

threshold can be found by minimizing the error probability, what can be

somewhat cumbersome. It is important to note that the thresholds vary with

the actual SNR, this will be clear through the next figures.

Examples of a 4-ASK and a BASK SLL Modulation

In Fig. 8.3, four beamformers generated through the solution of the

convex optimization problem (8.4) - (8.6) are depicted, w0, w1, w2 and w3.

They embed towards the communication receiver uc = −0.6 or θ = 126o the

symbols δ0 = B0(uc) =√10−20/10 = 0.1, δ1 = B1(uc) =

√10−23.1/10 = 0.07,

δ2 = B2(uc) =√10−27.96/10 = 0.04 and δ3 = B3(uc) =

√10−40/10 = 0.01 from

the constellation C = {Ci}3i=0, where Ci = δi, i = {0, . . . , 3}. We consider a

ULA consisting of M = 10 elements spaced by half a wavelength. The phase

profile in (8.4) is given by φ(ui) = 2πui. The region U in (8.4) is given by

U = [−0.1736, 0.1736], which corresponds to θ = [80o, 100o]. The region Uin (8.5) is given by U = {[−1,−0.4226], [0.4226, 1]}, which corresponds to

θ = {[0o, 65o], [115o, 180o]}. Leaving the regions u = [−0.4226,−0.1736] and

u = [0.1736, 0.4226], which correspond to θ = [65o, 80o] and θ = [100o, 115o] as

the transition interval. Parameter ǫ in (8.5) is set so as to lead to -20 dB in

the power pattern. We chose this 4-ASK constellation so as to be equal to the

values assigned in [71].

Fig. 8.4 shows the conditional density function of the received symbols

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

w0: SLL=-20 dB at u

c=-0.6

w1: SLL=-23.1 dB at u

c=-0.6

w2: SLL=-27.9 dB at u

c=-0.6

w3: SLL=-40 dB at u

c=-0.6

Comm. direction

Figure 8.3: Power patterns with embedded amplitude modulation at uc = −0.6using (8.4) - (8.6).

Z0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

prob

abili

ty d

ensi

ty fu

nctio

n

0

5

10

15

20

25

30SNR = 10 dB

pz|3

(Z)

pz|2

(Z)

pz|1

(Z)

pz|0

(Z)

3-rd symbol2-nd symbol1-st symbol0-th symbol

T2,3

T1,2

T0,1

T-1,0

Z0Z1 Z2 Z3

Figure 8.4: Conditional probability density function of the symbols transmittedusing a 4-ASK constellation embedded using the beamformers of Fig. 8.3 andSNR = 10 dB.

using the beamformers depicted in Fig. 8.3, as well as the decision threshold

in a scenario with SNR = 10 dB. Fig. 8.5 repeats Fig. 8.4 in a scenario with

SNR = 20 dB. Fig. 8.6 repeats Fig. 8.4 in a scenario with SNR = 5 dB. The

SNR in dB is defined as

SNR = 10 log10

[

Er|C0|2 + |C1|2 + |C2|2 + |C3|2

4σ2n

]

. (8.26)

We will now analyse the binary ASK constellation (BASK). From Fig.

8.3 we choose two beamformers: w0 and w3. We show these beamformers

DBD



Z0 0.05 0.1 0.15

prob

abili

ty d

ensi

ty fu

nctio

n

0

10

20

30

40

50

60

70SNR = 20 dB

pz|3

(Z)

pz|2

(Z)

pz|1

(Z)

pz|0

(Z)


T2,3

T1,2

T0,1

T-1,0

Z0 Z1Z2 Z3


Z0 0.05 0.1 0.15 0.2 0.25 0.3

prob

abili

ty d

ensi

ty fu

nctio

n

0

2

4

6

8

10

12

14

16

18SNR = 5 dB

pz|3

(Z)

pz|2

(Z)

pz|1

(Z)

pz|0

(Z)


T2,3

T1,2

T0,1

T-1,0

Z0Z2 Z3

Z1


separately in Fig. 8.7. They embed a binary constellation C = {C0, C1} =

{√10−20/10,

√10−40/10}, where

√10−20/10 is associated to bit “1” and

√10−40/10

corresponds to bit “0”. Fig. 8.8 shows the conditional density function of

the received symbols using the beamformers depicted in Fig. 8.7, as well as

the decision threshold in a scenario with SNR = 5 dB. Fig. 8.8 also shows

the transmitted symbol in the horizontal axis and their conditional density

function. Fig. 8.9 repeats Fig. 8.8 in a scenario with SNR = 10 dB. Fig. 8.10

repeats Fig. 8.8 in a scenario with SNR = 20 dB. From Figs. 8.8, 8.9 and

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

w0

w3

fidelity intervalComm. Direction

Maximum Sidelobe Level

Figure 8.7: Beamformers chosen to embed a binary amplitude constellationtowards uc = −0.6.

Z0 0.05 0.1 0.15 0.2 0.25

prob

abili

ty d

ensi

ty fu

nctio

n

0

2

4

6

8

10

12

14

16

18

20

SNR = 5 dB

pz|1

(Z)

pz|0

(Z)

1-st symbol0-thsymbolT

0,1

T-1,0

Z0 Z1

Figure 8.8: Conditional probability density function of the symbols transmittedusing a BASK constellation embedded using the beamformers of Fig. 8.7 andSNR = 5 dB.

8.10 we can see that the conditional probability densities of symbols “0” and

“1” are more distinguishable than the conditional probability densities of the

4-ASK case studied before.

Fig. 8.11 shows the symbol error probability, P (E), which is equal to

the BER, associated to the BASK signalling strategy and the symbol error

probability, P (E), associated to the 4-ASK signalling strategy discussed before.

The error probability, P (E), depicted in Fig. 8.11 was generated using (8.25),

where the decision thresholds were set at the intersection of the conditional

DBD



Z0 0.05 0.1 0.15 0.2 0.25 0.3

prob

abili

ty d

ensi

ty fu

nctio

n

0

5

10

15

20

25

30

35SNR = 10 dB

pz|1

(Z)

pz|0

(Z)

1-st symbol0-th symbolT

0,1

T-1,0

Z0 Z1

Figure 8.9: Conditional probability density function of the symbols transmittedusing a BASK constellation embedded using the beamformers of Fig. 8.7 andSNR = 10 dB.

Z0 0.05 0.1 0.15

prob

abili

ty d

ensi

ty fu

nctio

n

0

10

20

30

40

50

60

70

80

90SNR = 20 dB

pz|3

(Z)

pz|0

(Z)

3-rd symbol0-th symbolT

0,1

T-1,0

Z0Z1

Figure 8.10: Conditional probability density function of the symbols transmit-ted using a BASK constellation embedded using the beamformers of Fig. 8.7and SNR = 20 dB.

probabilities for each SNR. We can see that due to the radar constraints the

BASK signalling strategy leads to more operational bit error rates than the 4-

ASK signalling strategy and seems to be a more proper choice for dual-function

radar-communications platform.

DBD



SNR (dB)0 5 10 15 20 25 30

Pro

babi

lity

of s

ymbo

l err

or P

(E)

10-12

10-10

10-8

10-6

10-4

10-2

100

Analytical P(E) - 4ASKSimulated P(E) - 4ASKAnalytical P(E) - BASKSimulated P(E) - BASK

Figure 8.11: Symbol error probability for the BASK and 4-ASK constellationvs. SNR (dB).

Comments on the Security against Interception of the SLL Modulation

Now that we have chosen the BASK as a preferable SLL modulation

for dual-function radar-communications platform, let’s see how behaves the

angular BER. We need this angular BER behavior to attest the secure

communications property, that is, the property of leading to operational BER

only towards the communication direction. We will also need this angular BER

information in our discussion about robustness in Section 9.1.

In Fig. 8.12 we show the analytical angular BER associated to the chosen

signalling strategy. The decision threshold is the same for all directions. We

considered an AWGN channel for all angles and we scaled the noise power

σ2n = E [|n|2] so as to lead to

10 log10

(

Er|C0|2 + |C1|2

2σ2n

)

= 15 dB, u ∈ [−1, 1]. (8.27)

8.2 Sidelobe Phase Modulation

In this section, we will make an overview of the methods present in the

literature to embed phase modulation to the sidelobe of the radar transmit

beampattern. The methods so far described in the literature for this purpose

were proposed by the Villanova University Center for Advanced Communica-

tions research team. The method proposed in [3] achieves the design of multiple

transmit beampatterns which have exactly the same power radiation pattern

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

BE

R

10-7

10-6

10-5

10-4

10-3

10-2

10-1

Rx direction

Figure 8.12: Angular BER for a binary amplitude modulation using the methodof convex optimization described in [2].

and different phases towards a specific direction by exploring the beampattern

phase rotational invariance. The method of [4] proposes the design of beam-

patterns with different phases towards a specific direction by means of convex

optimization or, once again, by exploring the beampattern phase rotational

invariance.

The signalling strategies proposed in [3, 4] are non-coherent, therefore

it is necessary to work with phase differences. These methods suppose that

there are at least two orthogonal waveforms being transmitted at the same

time through different arrays with different beampatterns. The arrays are close

enough so that no extra phase difference is added. Each orthogonal waveform

is transmitted with a different beampattern that embeds a different phase

towards the communication receiver. The communication receiver will perform

matched filtering to each waveform and will compare the phase at the output

of each matched filter with a reference.

(a) Method of Phase Rotational Invariance

In the method of [3], N pairs of transmit orthogonal waveforms,

(sn(t), sn(t)), n = 1, . . . , N , are used to transmit N PSK symbols from a con-

stellation of size K, {φk}Kk=1, where the k-th symbol, φk, of the phase-rotation

dictionary embedded towards direction uc is given by

φk = ∠wHk s(uc)

wHk s(uc)

, (8.28)

where s(uc) is the steering vector pointed to the communication receiver.

DBD



Figure 8.13: Illustration of the signalling method of [3].

The same pairs of waveforms are used during all pulses while the pairs

of transmit beamforming weight vectors change from pulse to pulse based on

which entries of the constellation are to be embedded.

In this subsection we explain how the pair (N = 1) of transmit

beamforming weight vectors (wk, wk) is generated for embedding a certain

entry of the PSK constellation.

In Fig. 8.13 the signalling strategy is illustrated for an example of a phase-

rotation dictionary of size two (K = 2), that composes a BPSK constellation

with symbols φ1 and φ2. When symbol φ1 is triggered to be transmitted, the

pair of beamformers w1 and w1 are used for generating the beampattern of

two sensor arrays. Through the sensor array of beamformer w1 the waveform

s(t) is irradiated and through the sensor array of beamformer w1 the waveform

s(t) is irradiated. Equivalently, when symbol φ2 is triggered to be transmitted,

the pair of beamformers w2 and w2 are used for generating the beampattern of

two sensor arrays. Through the sensor array of beamformer w2 the waveform

s(t) is irradiated and through the sensor array of beamformer w2 the waveform

s(t) is irradiated.

The method proposed in [3] starts with a principal transmit beamforming

weight vector, w. which satisfies a certain desired transmit power radiation

pattern. TheM×1 principal weight vector can be used to generate a population

of 2(M−1) weight vectors of the same dimensionality which have the same

transmit power radiation pattern as that of w, [73]. The aforementioned

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

10lo

g 10(|

B(u

)|2)

-15

-10

-5

0

5

10

15

20Principal Beampattern

Figure 8.14: Principal beampattern.

population, denoted as W = {w1, . . . ,w2(M−1)}, can be obtained by viewing

the principal weight vector as a polynomial of order M − 1 with M − 1 roots

denoted as ri, i = 1, . . . , 2M − 1. Note that reflecting each root with respect

to the unit circle does not change the magnitude of the beampattern.

Comments on the Method of Rotational Invariance for Embedding Side-lobe Phase Modulation

Let’s observe an example of the phase rotational invariance method where

the principal power beampattern is depicted in Fig. 8.14. In the simulation

of Fig. 8.14 we used a transmit ULA of M = 10 antennas spaced half a

wavelength apart and assume that a communication receiver is located at

uc = 0.6428, or equivalently, θc = 40o. The beamformer is designed to focus

the transmit energy within the sector U = [−0.1736, 0.1736], which corresponds

to Θ = [80o, 100o] using spheroidal sequences [75]. Specifically, it is computed

as

w =

√

Et2(u1 + u2), (8.29)

where Et is the isotropically irradiated signal energy, normalized to be equal

to M , and u1 and u2 are the two principal eigenvectors of the matrix A given

byA =

∫

Us(u)sH(u)du. (8.30)

The principal weight vector of Fig. 8.14 is used to generate a population

of 2M−1 = 512 weight vectors which have exactly the same transmit power

patterns [73]. Fig. 8.15 shows the phase of each of the 512 beamformers towards

u = 0.6428.

DBD



30

210

60

240

90

270

120

300

150

330

180 0

Figure 8.15: Phase towards u = 0.6428 generated by all the 512 beamformers.

30

210

60

240

90

270

120

300

150

330

180 0

Figure 8.16: Phase difference towards u = 0.6428 generated by 262144 beam-former pairs.

From Fig. 8.15 we can note that if we could use coherent processing, this

formulation wouldn’t be appropriate for a QPSK modulation, for example, as

we can’t pick four beamformers that generate a QPSK modulation.

If we work with the phase difference between the pairs of beamformers

this situation improves. In Fig. 8.16 the population W is used to build 5122

pairs of vectors and the phase rotations associated with the communication

direction u = 0.6428 for the 262144 available pairs are depicted. Fig. 8.16

shows that the available phase-rotations cover the entire phase domain between

0o and 360o. This enables choosing a suitable phase rotation for any phase

constellation.

DBD



30

210

60

240

90

270

120

300

150

330

180 0

Figure 8.17: Phase towards u = 0.5 generated by all the 512 beamformers.

30

210

60

240

90

270

120

300

150

330

180 0

Figure 8.18: Phase difference towards u = 0.5 generated by 262144 beamformerpairs.

But as we can see in Figs. 8.17 and 8.18, this situation is different if

the communication receiver is located at uc = 0.5, or equivalently, θc = 60o.

In Fig. 8.17 the phase of all 512 beamformers towards u = 0.5 is depicted.

Fig. 8.18 shows the phase difference towards u = 0.5 for the same beamformer

pairs used to generate Fig. 8.16. We can see from Fig. 8.18 that not all phase

constellations can be used towards u = 0.5, as there are holes in the phase

diagram shown in Fig. 8.18.

With this simple example we can readily realize that not all phase

constellations are available towards all directions at the sidelobe. This fact

is not mentioned in [3], but the phase constellation has to be designed

DBD



thoughtfully, thinking that the receiver may be anywhere within the sidelobe.

Defining the beamformer pairs that will lead to the same phase constellation

within the sidelobe is a study that would have to be carried offline. This fact

also impacts directly on the storage capacity of the radar, as the pairs that

generate the desired phase constellation for one direction are not necessarily

the same that will generate the same phase constellation for other directions.

Another point that is not considered in [3] is the difficulty, probably

the impossibility of embedding phase modulation towards more than one

communication receiver. In [3] they consider that it would be possible to embed

phase modulation to L receivers located at the sidelobe. But as the population

W is limited, they don’t lead to enough flexibility for generating beamformers

that can embed phase modulation to more than one direction and deal with

the dynamics at the same time. This subject needs a careful study.

Bit Error Rate for the Binary Case


pulses, x(t, u), irradiated towards direction u = cos(θ), using the phase-

rotational invariance method to embed simultaneously N PSK symbols at the

sidelobe, is given by

x(t, u) =√

2Etejψ

∞∑

i=0

[N∑

n=1

(wHn,is(u)sn(t− iTPRI)

+ wHn,is(u)sn(t− iTPRI)

)], (8.31)

where the pair (wn,i, wn,i) ∈ {(wk, wk)}Kk=1 is one of the K possible

beamforming weighting vectors pairs, which embeds the n-th symbol during

the i-th pulse, s(u) is the array steering vector pointed at u, Et is the ar-

rays’ isotropically irradiated energy, ψ is the initial carrier phase, (sn(t), sn(t)),

n = 1, . . . , N , are N pairs of orthogonal radar waveforms, which are low-pass

pulses of duration τ , normalized to unitary energy and TPRI is the radar pulse

repetition interval.

At the communication receiver located at direction u of the transmit

beampattern, the outputs, rn[i] and rn[i], after demodulation, matched filtering

with the waveforms sn(t) and sn(t), n = 1, . . . , N , and sampling the outputs

at the point of maximum (see Fig. 8.19) are given by

rn[i] =√

ErejψwH

n,is(u) + nn, (8.32)

rn[i] =√

ErejψwH

n,is(u) + nn, (8.33)

DBD



X

1√2

r1[i]

rN [i]

r1[i]

rN [i]

y1[i] =

yN [i] =

rN [i]rN [i]

r1[i]r1[i]

6 ·

6 ·

φ1,i

φN,i

Compwiththres-holds

Compwiththres-holds

......

x(t, u) + n(t)

s1(τ − t)

s∗1(τ − t)

s∗N (τ − t)

sN (τ − t)

t = iTs + τ

Figure 8.19: Illustration of the rotational invariance-based non-coherent PSKreceiver.

where Er is the energy at the receiver side which was isotropically irradiated by

the transmitter, nn and nn, n = 1, . . . , N , are independent complex Gaussian

random variables with zero mean and variance σ2nn = σ2

nn = σ2n = N0.

The authors take the angle of the ratio yn[i],

zn[i] = ∠yn[i], (8.34)

where yn[i] is given by

yn[i] =rn[i]

rn[i], (8.35)

for comparison with the angular thresholds in order to define the output of

the n-th symbol of the i-th pulse, φn,i (see Fig. 8.19).

Thus, the decision variable, zn[i], due to the n-th symbol of the i-th

transmitted radar pulse at direction u, is given by

zn[i] = ∠

√Ere

jψwHn,is(u) + nn√

ErejψwHn,is(u) + nn

. (8.36)

Towards the receiver direction, uc, in the noise free environment we have

exactly φn,i = φn,i, φn,i ∈ {φk}Kk=1.

The non-coherent PSK receiver used in the rotational invariance-based

sidelobe modulation method is depicted in Fig. 8.19.

In Figs. 8.20 and 8.21 we show an example of a set of beamformers, w1,

w1, w2 and w2, used to embed a BPSK constellation φ1 = 0 corresponding to

bit “0” and φ2 = π corresponding to bit “1”, in tandem with two orthogonal

waveforms s(t) and s(t), using the method of rotational invariance described

in [3]. We can note from Figs. 8.20 and 8.21 that the four beamformers have

exactly the same power patterns as expected!

In Fig. 8.22 the phase difference towards uc of the beamformer pairs

(w1,w1) and (w2,w2) is depicted in polar coordinates. We can see that the

BPSK constellation is met sharply.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er P

atte

rn (

dB)

-15

-10

-5

0

5

10

15

20

w1

w1

Figure 8.20: Transmit power patterngenerated by the beamformers w0 andw0.

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er P

atte

rn (

dB)

-15

-10

-5

0

5

10

15

20

w2

w2

Figure 8.21: Transmit power patterngenerated by the beamformers w1 andw1.

30

210

60

240

90

270

120

300

150

330

180 0

Figure 8.22: Phase difference towards uc = 0.6428 of the beamformer pairs(w1,w1) and (w2,w2) in polar coordinates.

In this BPSK case, given that bit “0” is transmitted at the i-th radar

pulse, the ratio y1[i] = r1[i]/r1[i] is

y1[i] =

√Ere

jψwH0 s(uc)√

ErejψwH0 s(uc)

+ n0, (8.37)

y1[i] = 1 + n0, (8.38)

where n0 is non-Gaussian noise. Given that bit “1” is transmitted the ratio

DBD



SNR (dB)0 2 4 6 8 10 12

BE

R

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Figure 8.23: Bit error probability for the non-coherent BPSK vs. SNR (dB)using the phase rotational invariance method.

y1[i] = r1[i]/r1[i] is

y1[i] =

√Ere

jψwH1 s(uc)√

ErejψwH1 s(uc)

+ n1, (8.39)

y1[i] = −1 + n1, (8.40)

where n1 is non-Gaussian noise. Calling z1[i] = ℜ{y1[i]}, the authors decidefor bit “0” if z1[i] > 0 and decide for bit “1” if z1[i] ≤ 0. Fig. 8.23 shows the

simulated BER using the Monte Carlo method for the described non-coherent

BPSK system where the SNR in dB is defined as

SNR = 10 log10

[Er(|wH

0 s(uc)|2 + |wH0 s(uc)|2 + |wH

1 s(u)|2 + |wH1 s(u)|2)

4σ2n

]

.

(8.41)

Comments on the Security against Interception of the Phase RotationalInvariance Method

Fig 8.24 shows the phase difference of the beamformer pairs (w1,w1)

and (w2,w2) in the u-space for the considered non-coherent BPSK system. We

can note from Fig. 8.24 that towards the communication receiver the phase

difference is 180o and that this difference is repeated for many other directions

as well. This fact is reflected in the angular BER as can be noted from Fig.

8.25.

In Fig. 8.25, we simulated the transmission of 106 symbols for each angle

and guaranteed that at each angle the SNR was of 10 dB.

We can note that Fig. 8.25 is in agreement with Fig. 8.24 and that

DBD



the same BER obtained towards uc can also be obtained for many other

directions. We can note that the majority of the u-space is awarded with

operational BERs. This characteristic of the rotational invariance method is

not appropriate for secure communications, as it doesn’t lead to operational

BER only towards the communication receiver region.

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pha

se

-200

-150

-100

-50

0

50

100

150

180200

(w1, w1)

(w2, w2)

RXdirection

Figure 8.24: Phase difference in the u-space generated by the beamformer pairs(w1, w1) and (w2, w2).

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

BE

R

10-5

10-4

10-3

10-2

10-1

100

RXDirection

Figure 8.25: Angular BER for a BPSK modulation using the method ofrotational invariance described in [3].

(b) Method of Common Reference

The method of [3] uses a pair of orthogonal waveforms in tandem with

a pair of beamformers to non-coherently embed a PSK symbol. If there was

DBD



available, for example, four orthogonal waveforms, the method of [3] would use

two pairs of orthogonal waveforms in tandem with two pairs of beamformers

to transmit two PSK symbols during each pulse. What was noted in [4] is that

one can make it better. In that example of having four orthogonal waveforms,

one could use a common phase reference being transmitted by beamformer w0

thorough one of these orthogonal waveforms, then one could non-coherently

embed three PSK symbols during each pulse in the difference of the other

three beamformers used to transmit the other three orthogonal waveforms,

increasing the symbol rate.

In Fig. 8.26 the signalling strategy of [4] is illustrated for an example

where four orthogonal waveforms, s0(t), s1(t), s2(t) and s3(t), are available

for transmitting simultaneously through four independent array of sensors.

Each array is able to change its own beampattern from pulse to pulse. In

this example, we use a PSK constellation of size three with symbols φ1, φ2

and φ3. The three symbols are mapped to three pairs of beamformer, where

a common beamformer is used in all of the pairs, (w1,w0), (w2,w0) and

(w3,w0). Beamformer w0 is always used to generate the beampattern that will

irradiate waveform s0(t). The other three waveforms will have their radiation

patterns generated according to the sequence of symbols to be embedded.

In the example of Fig. 8.26, the sequence φ1, φ1 and φ3 is triggered to be

transmitted. Thus beamformer w1 is used to generate the radiation pattern

of waveforms s1(t) and s2(t) and beamformer w3 is used to generate the

radiation pattern of waveforms s3(t). If the signalling strategy used was of

[3], only two symbols could be embedded using the four orthogonal waveforms

simultaneously, instead of three.

In the method of [4], N orthogonal waveforms are used to simultaneously

embed N −1 PSK symbols. The PSK symbols belong to a constellation of size

K, where symbol φk ∈ {φi}Ki=1, embedded towards direction uc is given by

φk = ∠wHk s(uc)

wH0 s(uc)

. (8.42)

Also in [4], the authors suggest generating the beamformers using convex

optimization, similarly to the SLL formulation. In the optimization problem

formulation of [4], they suppose there is a reference beamformer w0 and the

other beamformers, wk, k = 1, . . . , K, will try to approach the beampattern

of w0 the best as they can while embedding a certain phase towards the

communication receiver direction, uc. One way to design wk is by minimizing

the norm of the difference between the two weight vectors, i.e., by minimizing

the deviation of wk from w0, while enforcing one linear constraint to satisfy

the phase requirements. This can be formulated as the following optimization

DBD



Figure 8.26: Illustration of the signalling method of [4] for the sequence ofsymbols φ1, φ2 and φ3.

problem

minwk ||w0 −wk||, (8.43)

s.t. wHk s(uc) = G0e

−jφk ,

where G0 = wH0 s(uc) and φk = {φ1, . . . , φK}. The optimization problem (8.43)

is convex and, therefore, can be efficiently solved using the interior point

methods [74]. Another advantage to this convex optimization-based design

compared to the phase rotational invariance design is that the desired phase

rotations are satisfied with equality. However, the obtained solutions yield

transmit weight vectors with very similar patterns to that of w0, but not

identical as it happens using the phase rotational invariance design.

Bit Error Rate for the Binary Case


pulses, x(t, u), irradiated towards direction u = cos(θ), using the method of [4]

to embed simultaneously N − 1 PSK symbols at the sidelobe, is given by

x(t, u) =√

2Etejψ

∞∑

i=0

(

wH0 s(u)s0(t− iTPRI) +

N−1∑

n=1

wHn,is(u)sn(t− iTPRI)

)

,

(8.44)

where w0 ∈ CM×1 is the reference beamformer which is always transmitted

in tandem with waveform s0(t). The beamformers wn,i ∈ {wk}Kk=1 embed the

N-1 symbols φn,i = wHn,is(u)/w

H0 s(u) ∈ {φk}Kk=1 during the transmission of

DBD



the i-th radar pulse. The array steering vector pointed at u is s(u), Et is

the isotropically irradiated signal energy by each array, ψ is the initial carrier

phase, sn(t), n = {0, . . . , N−1} are orthogonal waveforms, which are low-pass

pulses of duration τ , normalized to unitary energy and TPRI is the radar pulse

repetition interval.


beampattern, the outputs, rn[i], n = {0, . . . , N − 1} after demodulation,

matched filtering with the waveforms sn(t), n = {0, . . . , N − 1} and sampling

the N outputs at the point of maximum (see Fig. 8.27) are given by

r0[i] =√

ErejψwH

0 s(u) + n0, (8.45)

rn[i] =√

ErejψwH

n,is(u) + nn, n = {1, . . . , N − 1}, (8.46)

where Er is the energy, which was isotropically irradiated by the transmitter,

at the receiver side, considering the propagation attenuation effect, nn, n =

{0, . . . , N − 1} are independent complex Gaussian random variables with zero

mean and variance σ2nn = σ2

n = N0.

The authors take the angle of the ratio yn[i],

zn[i] = ∠yn[i], (8.47)

where yn[i] is given by

yn[i] =rn[i]

r0[i], n = 1, . . . , N − 1, (8.48)

for comparison with the angular thresholds in order to define the output of

the n-th symbol of the i-th pulse transmitted, φn,i (see Fig. 8.27).

Thus, the n-th decision variable due to the i-th transmitted radar pulse,

zn[i], n = {1, . . . , N − 1}, at direction u, is given by

zn[i] = ∠

√Ere

jψwHn,is(u) + nn√

ErejψwH0 s(u) + n0

. (8.49)

Towards the receiver direction, uc, in the noise free environment we have

exactly φn,i = φn,i, φn,i ∈ {φk}Kk=1.

The non-coherent PSK receiver used in the common reference-based

sidelobe modulation method is depicted in Fig. 8.27.

In Fig. 8.28 we show an example of a set of beamformers, w0, w1 and

w2, used in tandem with two orthogonal waveforms, s0(t) and s1(t), to embed

a BPSK constellation towards uc = −0.6, φ1 = 0 which corresponds to bit

“0” and φ2 = π which corresponds to bit “1”, using the signalling strategy

of [4]. The beamformers w1 and w2 were generated by solving the convex

DBD



r1[i]1√2

X

y1[i] =r1[i]r0[i]

yN−1[i] =

rN−1[i]r0[i]

6 ·

6 ·

φN−1,i

φ1,i

Comp

Comp

...

x(t, u) + n(t)s∗0(τ − t)

s∗1(τ − t)

rN−1[i]s∗N−1(τ − t)

r0[i]t = iTs + τ

Figure 8.27: Illustration of the common reference-based non-coherent PSKreceiver.

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Bea

mpa

ttern

(dB

)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

w1

w2

w0Comm.

Direction


Figure 8.28: Set of beamformers, w0, w1 and w2, used to embed a BPSKconstellation, φ1 = 0 and φ2 = π, using the signalling strategy of [4].

optimization problem of (8.43), where G0 = wH0 s(uc), φ1 = 0 and φ2 = π.

We can note from Fig. 8.28 that beamformer w1 led to the same beampattern

as w0 as expected, because the phase difference between them both is zero.

On the other hand, beamformer w2, led to a similar, but not exactly the

same beampattern as w0. In Fig. 8.29 the phase difference towards uc of the

beamformer pairs (w1,w0) and (w2,w0) is depicted in polar coordinates. We

can see that the BPSK constellation is met sharply.

Note that the method of [4] does not explicitly states the formulation of

maximum allowable sidelobe level, therefore, not necessarily, the beamformers

will cope with this operational restriction, as can be seen in Fig. 8.28.

In this BPSK case, given that bit “0” is transmitted, the ratio y1[i] =

DBD



30

210

60

240

90

270

120

300

150

330

180 0

Figure 8.29: Phase difference towards uc of the beamformer pairs (w1,w0) and(w2,w0).

r1[i]/r0[i] is

y1[i] =

√Ere

jψwH1 s(uc)√

ErejψwH0 s(uc)

+ nb0 , (8.50)

y1[i] = 1 + nb0 , (8.51)

where nb0 is non-Gaussian noise. Given that bit “1” is transmitted the ratio

y1[i] = r1[i]/r0[i] is

y1[i] =

√Ere

jψwH2 s(uc)√

ErejψwH0 s(uc)

+ nb1 , (8.52)

y1[i] = −1 + nb1 , (8.53)

where nb1 is non-Gaussian noise. Calling z1[i] = ℜ{y1[i]}, the authors decidefor bit “0” if z1[i] > 0 and decide for bit “1” if z1[i] ≤ 0. Fig. 8.23 shows the

simulated BER for the described non-coherent BPSK system, where the SNR

in dB is defined as

SNR = 10 log10

[

Er(|wH

0 s(u)|2 + |wH1 s(u)|2 + |wH

2 s(u)|2)

3σ2n

]

. (8.54)

Comments on the Security against Interception of the Common ReferenceMethod

Fig. 8.31 shows the phase difference of the beamformer pairs (w1,w0) and

(w2,w0) in the u-space. We can note from Fig. 8.31 that the most expressive

DBD



SNR0 2 4 6 8 10 12

BE

R

10-6

10-5

10-4

10-3

10-2

10-1

100

Figure 8.30: Bit error probability for the non-coherent BPSK vs. SNR (dB)using the common reference method.

phase changes happen only around the communication direction.

In Fig. 8.32 it is depicted the angular BER for a BPSK modulation

embedded towards a communication receiver located at uc = −0.6 using the

method of [4]. The beamformers are the same as depicted in Fig. 8.31. In

Fig. 8.32, we simulated the transmission of 106 symbols for each angle and

guaranteed that at each angle the SNR was of 10 dB.

From Figs. 8.31 and 8.32 we can note that a highly secure communication

system can be achieved using this method, differently than the rotational-

invariance method of [3].

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

phas

e

-150

-100

-50

0

50

100

150phase difference due to beamformers pair (w

1, w

0)

phase difference due to beamformers pair (w2, w

0)

communication receiver direction

Figure 8.31: Phase difference of the beamformer pairs (w1,w0) and (w2,w0) inthe u-space.

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

BE

R

10-5

10-4

10-3

10-2

10-1

100

RX.direction

Figure 8.32: BER in the u-space for the described BPSK system using thebeamformer pairs (w1,w0) and (w2,w0).

DBD


9Proposed Radar-Embedded Sidelobe Modulation

In this chapter, we present our proposed methods for embedding sidelobe

modulation to the radar transmit beampattern. In the former chapter we have

called the reader’s attention to the difficulty of having enough flexibility to

pursue the communication receiver movement in a real-time application. We

have also alerted about the need of secure communications and introduced

the concept of robustness within this context. In Section 9.1 we deepen the

discussion about robustness to emphasize the need of methods that deal with

this issue. In this chapter, we focus on two, not yet explored, aspects of the

dual-function radar-communications: robustness and real-time tracking

ability.

In this thesis, we propose radar-embedded robust sidelobe modulation

methods for dual-function radar-communication systems. One of our proposed

techniques is based on quadratic and linear constrained optimization design,

which has globally optimal closed-form solution. This proposed method gener-

ates transmit beampatterns that satisfactorily match a given transmit profile

(with high fidelity adjustment at the mainlobe), where each beampattern em-

beds a different symbol towards the communication receiver direction and sus-

tains it over a small angular region. This sustaining ability makes the proposed

method robust against small errors over the expected position of the commu-

nication receiver. This characteristic provides more flexibility to the system

and allows real-time processing to cope more easily with moving communica-

tion platforms, since operational BERs are achieved in a small region, rather

than in a single point in the u-space. Part of this study resulted in the paper

[7].

In this thesis, we also propose an iterative version of this proposed

method. We apply the constrained gradient descent method for solving the

constrained optimization problem iteratively. This new approach converges

to the optimal solution. The recursive formula is very interesting within the

DFRC context, because it it possible to come up with a stop criteria that

are directly related to the radar operation, rather than minimization of the

squared error. We derive simple equations for updating the beampatterns for

following a moving communication receiver platform for both the closed form

DBD


Chapter 9. Proposed Radar-Embedded Sidelobe Modulation 132

and the iterative version of this proposed method.

We also propose an alternative method for radar-embedded robust side-

lobe modulation for a dual-function radar-communication system. We derive

this alternative method by modifying the robust optimization problem, in order

to handle differently the mainlobe adjustment requirement. The proposed for-

mulation is based on eigenvector, point and first derivative constraints. Though

this solution is different, our alternative proposed method is also able to gen-

erate transmit beampatterns that match satisfactorily a given transmit profile

(with high fidelity adjustment at the mainlobe), where each beampattern em-

beds a different symbol towards the communication receiver direction and sus-

tains it over a small angular region. We derive its closed form solution and we

simplify it through mathematical manipulation and eigenspectrum analysis.

We also derive simple equations for updating the beampatterns for following a

moving communication receiver platform. These procedures make the proposed

alternative solution also very interesting and suitable for real-time processing.

Part of this study resulted in the paper [8].

All proposed methods are applicable to both sidelobe amplitude and

phase modulation. Referring to phase modulation, we propose a new non-

coherent signalling strategy within the DFRC context. We also derive the bit

error rate expression for the proposed phase modulation signalling strategy

when two arbitrary symbols are transmitted. To the best of our knowledge,

this expression is not found in the literature, but it is necessary for computing

the angular BER figures, therefore, we derive it here.

In Chapter 10 we present some computer simulations and comparisons of

the methods. In Section 10.8 we discuss the effects of sidelobe modulation in

clutter mitigation techniques and in Section 10.9 we report some conclusions.

9.1 Considerations About Robustness

In this section, we will make some comments about the robustness

of the methods seen so far. As mentioned before, the secure behavior of

the radar-embedded communication system based on sidelobe modulation

is intrinsic to its formulation. The preassignment of the receiver direction

provides operational BER only towards this direction (except for the phase

modulation using phase rotational invariance method), but this inherent

advantage may become a major disadvantage if the real receiver position

doesn’t match exactly the preassigned one.

At this point, we use the one-round trip range equation to relate

deterministically the received power at the communication receiver to the

transmitted power in terms of a variety of system design parameters. After

DBD



that, we consider the random behavior inherent in communication signals.

First, assume an isotropic radiating element transmitting a waveform of

power Pt Watts into a lossless medium. Thus, the power density at a range R,

S(R), is the transmitted power Pt divided by the surface area of a sphere of

radius R,S(R) =

Pt4πR2

W/m2. (9.1)

But in practice, real radars use antenna arrays to focus the outgoing energy,

thus the power density incident upon a communication receiver located towards

u at range R from the radar, S(R, u), is given by

S(R, u) =Gt|B(u)|2Pt

4πR2W/m2, (9.2)

where Gt is the transmitting array gain, which is the ratio between the

maximum power density to the isotropic density, and B(u) is the normalized

beampattern. The quantity GtB(u) defines the array gain towards u, G(u),

which is the ratio between the power density towards u to the isotropic density.

If the effective aperture size of the communication receiver antenna is Ae

m2, the total power collected by the receiving antenna, located towards u at

range R from the transmitter, Pr(R, u), will be

Pr(R, u) =AeGt|B(u)|2Pt

4πR2W, (9.3)

but the effective aperture of the receiver antenna is related to its gain, Gr, and

operating wavelength, λ, according to

Ae =λ2Gr

4π, (9.4)

thus, substituting (9.4) into (9.3) we have

Pr(R, u) =GrGt|B(u)|2λ2Pt

(4π)2R2W. (9.5)

We assume that the communication receiver antenna’s direction of maximum

gain is aligned with the radar, therefore, Gr, is the ratio between the maximum

power density of the receiver antenna’s radiation pattern to the isotropic

density.

Various additional loss and gain factors are customarily added to (9.5) to

account for a variety of additional considerations. For example, losses incurred

in various components such as the duplexers, power dividers, waveguide and

radome (a protective covering over the antenna), and the propagation effects

not found in free space propagation can be lumped into a system loss factor,

Ls, that reduces the received power. Considering the system loss factor, Ls,

(9.5) becomes

DBD



Pr(R, u) =GrGt|B(u)|2λ2Pt

(4π)2R2LsW. (9.6)

The SNR at the input of the communication receiver at range R from the

transmitter is given by dividing (9.6) by the receiver’s noise power, kT0BF ,

SNR =Pr(R, u)

kT0BF, (9.7)

where k = 1.38 × 10−23 J/K is the Boltzmann’s constant, T0 = 290 K is

the standard noise temperature, F is the receiver noise figure and B is the

communication receiver’s input bandwidth in Hz, which is assumed to be

matched to the particular radar waveform bandwidth that is being transmitted.

Multiplying and dividing (9.7) by the pulse duration, τ , we have the SNR

written in function of the ratio of the signal and noise energy,

SNR =Pr(R, u)τ

kT0FBτ=

Er(R, u)

kT0F (Bτ), (9.8)

where Er(R, u) is the energy collected by the receiver’s antenna located at

range R and direction u relative to the radar. The minimum value for the

product Bτ , in order to avoid intersymbol interference is one. A typical value

of Bτ for a pulse with roll-off is of 1.4.

But as we are considering a communication context, the SNR is then the

ratio between the expected value of the energy relative to all the K symbols

being transmitted and the receiver noise power kT0BF ,

SNR =E [Er(R, u)]

kT0F (Bτ). (9.9)

Considering that all communication symbols are equally likely of being

transmitted the SNR at the communication receiver is thus given by

SNR =GrGtPtτλ

2

KkT0BτF (4π)2R2Ls

K∑

k=1

|Bk(u)|2, (9.10)

where Bk(u) is the normalized beamppattern towards u.

Let’s define Rmax as the maximum range at which the communication

receiver can achieve its minimum operational SNR, SNRmin. Any R greater

than Rmax leads to a lower SNR than SNRmin, which in its turn makes

communication impractical. From (9.10) we can find the maximum range,

Rmax, at which the radar can communicate with the communication receiver

as a function of the SNRmin,

Rmax =

√√√√ GrGtPtτλ2

KkT0BτF (4π)2SNRminLs

K∑

k=1

|Bk(u)|2. (9.11)

DBD



SNRmin

(dB)0 2 4 6 8 10 12 14 16 18 20

Max

imum

ran

ge (

Km

)

100

200

300

400

500

600

700

800

900

1000

1100

Figure 9.1: Maximum range as a function of SNRmin for the first example.

Let’s use equation (9.11) for a simple, yet representative example. Con-

sider a radar with transmission power of Pt = 80 W, transmit gain of

Gt = 20 dB, waveform with a 2 MHz bandwidth and central frequency of

f = 9.375 GHz. We choose the BASK SLL modulation of [3] that uses the

two beampatterns depicted in Fig. 8.7 for embedding the binary constella-

tion C = {√10−20/10,

√10−40/10}, where

√10−20/10 is associated to bit “1” and√

10−40/10 corresponds to bit “0” towards uc = −0.6. The communication re-

ceiver has a noise figure of F = 5 dB, antenna gain of Gr = 10 dB and the

systems losses are Ls = 5 dB. Fig. 9.1 shows the maximum range as a function

of SNRmin at the input of the communication receiver.

According to Fig. 8.11 a SNR of 15 dB is operational for this signaling

strategy as it leads to a BER of 3.17 × 10−7. For this example the maximum

range at which the radar can communicate with the communication receiver

for a SNR at the input of the communication receiver of 15 dB is of 180 Km.

Now let’s assume that the communication receiver is in a corvette

moving further from the radar mainbeam with a cruising speed of 17 knots or

equivalently 8.75 m/s in a direction tangential to the circumference of radius

180 Km centered at the radar as depicted in Fig. 9.2. In 10 minutes the relative

angle between the radar and the communication receiver would have increased

of ∆u = −0.025. Checking the angular BER in Fig. 8.12 we can see that the

BER would go from 3.17×10−7 to 1.1×10−3 in only 10 minutes. In 22 minutes

∆u = −0.05 (∆θ = 3.67o) and the BER goes to 0.27. This BER dynamic is

illustrated in a zoom of Fig. 8.12, Fig. 9.3.

Now let’s see another example, this time the BPSK embedded phase

modulation using the signalling and convex optimization method of [4]. In the

DBD



Figure 9.2: Moving communication receiver.

u-0.7 -0.68 -0.66 -0.64 -0.62 -0.6 -0.58 -0.56 -0.54 -0.52 -0.5

BE

R

10-8

10-6

10-4

10-2

10 minutes ∆u

20 minutes ∆u

Figure 9.3: Zoom of BASK angular BER showing the BER degradationaccording to communication receiver motion for the first example for a SNRof 15 dB.

following simulation we used the beampatterns as in Fig. 8.28 that embed

towards uc = −0.6 the phase constellation φ1 = 0 which corresponds to bit

“0” and φ2 = π which corresponds to bit “1”. The angular BER is depicted in

Fig. 8.32. This signalling method uses two different arrays to simultaneously

transmit two waveforms, so we will assume that each array has a transmit

power Pt of 80/2 = 40 W. We consider Gt = 20 dB, the waveforms have a 2

MHz bandwidth and central frequency of f = 9.375 GHz. The communication

receiver has a noise figure of F = 5 dB, antenna gain of Gr = 10 dB and the

systems losses is Ls = 5 dB. Fig. 9.4 shows the maximum range as a function

of SNRmin at the input of the communication receiver.

According to Fig. 8.23 a SNR of 10 dB is operational for this signalling

strategy as it leads to a BER of 2.25 × 10−5. For this example the maximum

range at which the radar can communicate with the communication receiver

for a SNR at the input of the communication receiver of 10 dB is of 205 Km.

We will use the same receiver movement information as in the first

DBD



SNRmin

(dB)0 5 10 15

Max

imum

ran

ge (

Km

)

100

200

300

400

500

600

700

Figure 9.4: Maximum range as a function of SNRmin for the second example.

example. After 12 minutes the relative angle between the radar and the

communication receiver would have increased of ∆u = −0.025. Checking the

angular BER in Fig. 8.32 we can see that the BER would go from 2.25× 10−5

to 0.009 in only 12 minutes. In 25 minutes ∆u = −0.05 and the BER goes to

0.3341. This BER dynamic is illustrated in a zoom of Fig. 8.32, Fig. 9.5.

The relative movement assumed within this robustness analysis is just

a simple and particular example. We assumed a fixed landbased radar and a

ship movement, which is considerable slower than a flying target for example.

Depending on the scenario, the time for the platform of the communication

receiver to course this relative angular displacement may be considerably

smaller.

These examples illustrate our idea of the necessity of robust and

flexible methods for embedding SLL modulation to dual-function radar-

communications platforms. We won’t even consider here the case of a rotating

radar. A rotating radar of 10 rpm would go from uc = -0.6 to -0.65 in mili-

seconds! But we won’t address this problem here.

9.2 Proposed Robust Real-Time Radar-Embedded Sidelobe Modulation- Method 1

In this section, we detail our first proposed robust radar-embedded

sidelobe modulation method. We propose a new formulation of the sidelobe

modulation problem for ULAs, which allows a simple, robust and closed

form solution. The closed form solution can be achieved iteratively, which

is developed in Subsection 9.2.2. We also derive simple update equations for

pursuing a moving communication receiver for the proposed method 1 in its

DBD



u-0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45

BE

R

10-5

10-4

10-3

10-2

10-1

100

12 minutes ∆u

25 minutes ∆u

Figure 9.5: Zoom of non-coherent BPSK angular BER showing the BER de-gradation according to communication receiver motion for the second examplefor a SNR of 10 dB.

closed form and recursive version.

First, let us state our assumptions:

– The radar designers have already developed a transmit beampattern

profile which meets the operational radar needs. The original transmit

beampattern, Bo(u), is described by the beamformer, wo, producing the

beampattern Bo(u) = wHo s(u), u ∈ [−1, 1];

– The radar primary function takes place at the mainlobe of the trans-

mit beampattern, u ∈ U , U = [−ud, ud]. In order to prevent perform-

ance degradation of the primary radar task, while embedding sidelobe

modulation, we have to assure that the mainbeam of all beampatterns

under design will be in good agreement with the original beampattern,

Bo(u).

– The receiver communication direction, uc, is known a priori.

Now, let us define the difference beamforming vector w, as

w = wo −wk, (9.12)

where wo is the given beamforming vector that produces the specific radar

transmit output power profile and wk is the beamforming weighting vector

under design, which will embed the k-th communication symbol towards uc.

Clearly, w depends on k. But this dependence will not be explicit in the

notation for simplicity. We will limit the optimization formulation to finding

the optimal w, named wop, then wopk is obtained by

DBD



wopk = wo −wop. (9.13)

The original beampattern, Bo(u), at the mainlobe spatial sector, U =

[−ud, ud], is given by

Bo(u) = wHo s(u), u ∈ U. (9.14)

One main design requirement of our beamforming vector, wk, is that in the

mainlobe area, u ∈ U , the total mean squared error, e, should be held under

a certain baseline, δ,

e =

∫

u∈U|wH

o s(u)−wHk s(u)|2du

=

∫

u∈U|wHs(u)|2du ≤ δ. (9.15)

We can rewrite (9.15) as [10]

e = wHQw, (9.16)

where Q ∈ CM×M is the hermitian symmetric positive semi-definite matrix,

given byQ ,

∫

u∈Us(u)sH(u)du. (9.17)

We can note that Q in (9.17) does not depend on the beampattern shape, but

only on the constraint region and the steering vector, s(u). The mn-th element

of Q, using d = λc/2, is given by [10]

[Q]mn =

∫ ud

−udejπu(m−n)du, (9.18)

[Q]mn =2 sin [πud(m− n)]

(m− n). (9.19)

According to (9.19), if we consider the squared error of the entire visible region,

then ud = 1 andQ = 2πIM , thus, minimizing ||w||2 is equivalent to minimizing

the total squared error. Now we can formulate the optimization problem on

w by stating that, wop is the difference beamforming vector that satisfies the

following

wop = minw

||w||2, (9.20)

s.t.

{

wHQw ≤ δ

wHs(uc) = dk,

where dk = wHo s(uc)− Ck and Ck is the k-th symbol of the constelation C.

In order to be more robust against small deviations in uc, we impose the

DBD



first derivative null constraint writing the optimization problem in (9.20) as

wop = minw

||w||2, (9.21)

s.t.

wHQw ≤ δ

wHs(uc) = dk

wHs′(uc) = b

,

where s′(uc) is the derivative of s(u) with respect to u evaluated in uc and

b = wHo s

′(uc).

By changing the ellipsoid constraint inequality into an equality, as

wHQw = δ, we can solve problem (9.21) by using the Lagrange multipliers

methodology based on the real valued function

L = wHw + λ1(wHQw − δ) (9.22)

+ 2ℜ{λ2wHs(uc)− dk}+ 2ℜ{λ3wHs′(uc)− b},

where λ1, λ2 and λ3 are the Lagrange multipliers. Employing the complex

gradient approach of [76] we have that the gradient of (9.22) with respect to

the conjugate of w, w∗, is

1

2∇w∗L = w + λ1Qw + λ2s(uc) + λ3s

′(uc), (9.23)

or equivalently, 1

2∇w∗L = w + λ1Qw + Sλ, (9.24)

where

S = [s(uc), s′(uc)] ∈ C

M×2, (9.25)

λ = [λ2, λ3]T ∈ C

2×1. (9.26)

The stationary point is given by nulling (9.24), thus

w = −(IM + λ1Q)−1Sλ. (9.27)

Employing the equality constraints in (9.21) it can be shown that

λ = −[SH(IM + λ1Q)−1S]−1fk, (9.28)

where fk is given by

fk =

[

d∗kb∗

]

∈ C2×1. (9.29)

Substituting (9.28) into (9.27) we have that

wop = (IM + λ1Q)−1S[SH(IM + λ1Q)−1S]−1fk. (9.30)

DBD



The value of λ1 is set so as to comply with the constraint wHQw ≤ δ. In

order to keep the consistency constraint of the optimization problem, we

must have λ1 ∈ [0,+∞) according to the Karush-Kuhn-Tucker condition

[74]. Furthermore, since Q is positive semi-definite, its eigenvalues are all non

negative, and thus, the eigenvalues of (IM +λ1Q) are all positive (in fact, they

are not less than one). This guarantees that matrix IM +λ1Q in (9.30) has an

inverse for all λ1 ≥ 0. Let λ1 be given by

λ1 =α

1− α, α ∈ [0, 1), (9.31)

where α is the adjustment parameter. Its physical interpretation will be

developed in the next subsection. Substituting (9.31) into (9.30), after some

manipulation we have that, for a given α, the optimal solution is given by

wop = Q−1(α)S(SHQ−1(α)S)−1fk, (9.32)

whereQ(α) = (1− α)IM + αQ, α ∈ [0, 1]. (9.33)

(a) Another Interpretation of the Proposed Technique

The optimal solution given in (9.32) can be found also by minimizing the

intuitive convex combination of two functions

wop = minw

(1− α)||w||2 + αwQw, (9.34)

s.t.

{

wHs(uc) = dk

wHs′(uc) = b,

where α ∈ [0, 1), ||w||2 is the total squared difference between the beampattern

under design and the original one and wQw is the squared difference between

the beampattern under design and the original one within the mainbeam

sector. Employing the Lagrange multipliers methodology we have

L = (1− α)wHw + αwHQw (9.35)

+ 2ℜ{λ2wHs(uc)− dk}+ 2ℜ{λ3wHs′(uc)− b},

differentiating (9.35) with respect to the conjugate of w, w∗, we have

1

2∇w∗L = (1− α)w + αQw + Sλ, (9.36)

where S is defined in (9.25) and λ is defined in (9.26). Nulling (9.36) to find

the stationary point we achieve

wop = −[(1 − α)IM + αQ]−1Sλ. (9.37)

DBD



Number of complex products Number of complex additions(1− α)IM + αQ︸︷︷︸

Q

M2 M + 1

Q−1

︸︷︷︸

Qin

O(M3) O(M3)

SHQinS︸︷︷︸

A

2M2 + 4M 2M2 + 2M − 4

(A)−1fk

︸︷︷︸

−λ

10 3

−QinSλ M2 + 2M M2

Total O(M3) + 4M2 + 6M + 10 O(M3) + 3M2 + 3M

Table 9.1: Complex operations of the proposed method 1 with the derivativenull constraint.

Applying the constraints to (9.37) we derive that

λ = −{SH [(1− α)IM + αQ]S}−1fk, (9.38)

which leads, for a given α, to the same optimal solution (9.32).

This interpretation is interesting to clarify the physical aspect of para-

meter α. We can see α as an adjustment factor that composes the final min-

imization solution. As we increase α, we give emphasis on the constraint that

controls the error at the mainbeam, as we decrease α, we relax the mainbeam

fit constraint and emphasize the total error minimization. Sometimes, if we

are too strict in the mainbeam fit, we can have the undesirable effect of high

sidelobes, but, by understanding the role of α we can adjust it to have the best

tradeoff between the mainbeam fit and sidelobe behavior. From the derivation

above, for a given threshold, δ, or equivalently, for a given α, the solution to

(9.32) is unique and optimal, implying that the solution to (9.32) achieves the

desired mainbeam fit with minimum total squared error.

The number of complex multiplications and additions necessary to com-

pute wop given α is detailed in Table 9.1.

If the communication receiver is moving in the u-space and we have

its displacement information, ∆u, we can compute the new beamforming

weighting vector wopk (u+∆u). Let’s call wop by wop(u), S by Su and fk by fk,u

in (9.32). Thus we can compute wop(u+∆u) by updating Su and fk,u as

Su+∆u = Su ⊙ S∆u, (9.39)

fk,u+∆u =

(

wHo [s(u)⊙ s(∆u)]− dk

wHo [s

′(u)⊙ s(∆u)]

)

, (9.40)

and substituting them into equation (9.32). Symbol⊙ stands for the Hadamard

DBD



element-wise product. Therefore the updated wopk (u+∆u) is given by

wopk (u+∆u) = wo − Q−1(α)Su+∆u(S

Hu+∆uQ

−1(α)Su+∆u)−1fk,u+∆u. (9.41)

(b) Recursive Version of the Proposed Method 1

In this section, we propose a constrained gradient-descent technique for

solving the robust radar embedded sidelobe modulation optimization problem

(9.21). The proposed recursive method converges to the optimal solution. It

is also very flexible in a sense that it is possible to add stop criteria that

are related to radar metrics other than minimization of squared errors. This

flexibility is very convenient in a DFRC context.

We will work with the simpler optimization formulation of (9.34), as it

was shown in Section 9.2 that, for the same α, (9.34) and (9.21) lead to the

same solution. The solution, wop of (9.34), can be found iteratively through

the update equation of the steepest descent methodology

w(n+ 1) = w(n)− µ∇w∗L, (9.42)

where 1

2∇w∗L = (1− α)w + αQw + Sλ, (9.43)

λ ∈ C2×1 is the vector λ = [λ2, λ3]

T composed of the Lagrange multipliers λ2

and λ3 explicit in equation (9.35).

Substituting (9.43) into (9.42) and incorporating the factor 1/2 to the

step, µ, we have that

w(n+ 1) = w(n)− µ [(1− α)w(n) + αQw(n) + Sλ] ,

= w(n)− µ(

Qw(n) + Sλ)

, (9.44)

where Q = (1 − α)IM + αQ. The Lagrange multipliers are chosen so that

w(n+ 1) satisfies the constraint SHw = fk,

SHw(n+ 1) = SHw(n)− µ(

SHQw(n) + SHSλ)

. (9.45)

Inspired by [19], we won’t consider SHw(n) = f in (9.45) (as it would be if

w(n) satisfied the constraint precisely), in order to correct any small errors

due to arithmetic inaccuracy, preventing their accumulation [19]. Thus, from

(9.45), λ is given by

λ = −(SHS)−1SHQw(n) +(SHS)−1

µ

(SHw(n)− fk

). (9.46)

DBD



Substituting (9.46) into (9.44) we have the recursive expression for w,

w(n+ 1) = w(n)− µ(I− S(SHS)−1SH

)Qw(n)

+ S(SHS)−1(fk − SHw(n)

). (9.47)

Again inspired by [19], defining

vk , S(SHS)−1fk ∈ CM×1, (9.48)

P , I− S(SHS)−1SH ∈ CM×M , (9.49)

and substituting into (9.47), we have that

w(n+ 1) = P(

I− µQ)

w(n) + vk, (9.50)

where w(0) , vk and 0 < µ < 2

3tr(Q)in order to assure convergence.

If the communication receiver is moving in the u-space and we have its

displacement information, ∆u, we can modify the update equation (9.50) in

order to follow the communication receiver, w(n + 1, u + ∆u). Let’s make

explicit the dependence on u by adding a subindex u to the quantities that are

dependent on the direction of the communication receiver, Su, fk,u, vk,u and

Pu. According to the communication receiver displacement ∆u, Su+∆u and

fk,u+∆ are updated as in (9.39) and (9.40) respectively. The quantities vk,u+∆u

and Pu+∆u are updated as

vk,u+∆u = Su+∆u(SHu+∆uSu+∆u)

−1fk,u+∆u, (9.51)

Pu+∆u = I− Su+∆u(SHu+∆uSu+∆u)

−1SHu+∆u. (9.52)

Substituting (9.51) and (9.52) into (9.50) we have that w(n + 1, u + ∆u) is

given by

w(n+ 1, u+∆u) = Pu+∆u

(

I− µQ)

w(n, u+∆u) + vk,u+∆u. (9.53)


pute wop given α using the recursive version of method 1 is detailed in Table

9.2.

DBD



Number of complex products Number of complex additionsInitial procedures

(SHS)−1

︸︷︷︸

S

4M + 6 4M − 3

SSfk︸︷︷︸

vk

8 4

IM − SSSH︸︷︷︸

P

2M2 + 4M M2 + 3M

(1− α)IM + αQ︸︷︷︸

Q

M2 M + 1

Start of a Nit iterations loop

(IM − µQ)w(n)︸︷︷︸

wµ(n)

2M2 M2

Pwµ(n) + vk M2 M2

Total 3M2 + 8M + 14 + 3M2Nit M2 + 8M + 2 + 2M2Nit

Table 9.2: Complex operations of the recursive version of the proposed method1 with the derivative null constraint.

9.3 Proposed Robust Real-Time Radar-Embedded Sidelobe Modulation- Method 2

In this section we propose another approach to the robust radar-

embedded sidelobe modulation optimization problem of (9.21). By means of

null eigenvalues constrained optimization design, we derive a solution, which is

also able to generate transmit beampatterns that match a given mainbeam out-

put power profile and embed sidelobe modulation towards the communication

receiver direction, sustaining the same value embedded over a small angular

interval. We simplify the proposed solution, reducing significantly the number

of necessary complex operations, thus, turning this proposed method also well

suited for online adaptive processing. We also derive simple update equations

for pursuing a moving communication receiver.

We propose another approach to deal with the requirement of constrain-

ing the error in the mainlobe, wHQw. Since

wHQw =

M∑

i=1

λi|wHui|2, (9.54)

where λi ≥ 0, i = 1, . . . ,M , are the eigenvalues of Q and ui is the eigenvector

of Q associated to its i-th eigenvalue, we can force the reduction of the squared

error, wHQw, by canceling the contribution of some eigenvectors of Q. We do

this by imposing null eigenvector constraints as

wHui = 0, i = 1, . . . , Ne, (9.55)

DBD



where {ui}, i = 1, . . . , Ne, are the eigenvectors of Q associated to its Ne

largest eigenvalues. We choose Ne to correspond to the number of significant

eigenvalues of Q. As we increase Ne, we reduce the squared error in (9.54), but

we also reduce the number of available degrees of freedom left for imposing the

sidelobe communication modulation. The value of Ne must be properly chosen

to adequately suit both primary and secondary radar functions. The proposed

optimization problem is, thus, given by

wop = minw

||w||2, subject to CHw = g, (9.56)

where C is given byC =

[

U... S

]

, (9.57)

where S is defined in (9.25), U ∈ CM×Ne is the matrix whose columns are the

Ne eigenvectors of Q associated to its Ne largest eigenvalues and g ∈ CNe+2×1

is given by g = [01×Ne , fTk ]T , where fk is defined in (9.29). Finally, for a given

Ne, the solution of (9.56) is given by

wop = C(CHC)−1g. (9.58)

Therefore, w∗k is given straightforwardly by

wopk = wo −wop. (9.59)

The proposed approach can be shown to be a suboptimal solution of

(9.21) in a certain sense since, for the same total squared error in the mainlobe

wHQw = δ, the beampattern, w, which is the solution of (9.21) is the one

that results in the minimum total error.

(a) Simplifications and Receiver Pursuit

In this section, we show how to simplify (9.58), avoiding the computation

of the inverse of a full matrix CHC ∈ C(Ne+2)×(Ne+2).

First, we can write CHC as

CHC =

[

UHU UHS

SHU SHS

]

=

[

INe A

AH D

]

, (9.60)

where UHU = INe because the columns of U are eigenvectors of Q, A ∈ CNe×2

and D ∈ C2×2. We can use the inversion property of partitioned matrices [10]

and write the inverse of CHC as

(CHC)−1 =

[

B−1 −B−1AD−1

−F−1AH F−1

]

, (9.61)

DBD



where B and F are given respectively by

B = INe −AD−1AH ∈ CNe×Ne, (9.62)

F = D−AHA ∈ C2×2. (9.63)

At this point, we have the inversion of matrices F and D that are of size 2×2,

which can be inverted efficiently with only a few complex product operations.

But we still have to invert matrix B, that is of size Ne×Ne. In order to simplify

B−1, we can apply the matrix inversion lemma [10] to (9.62) and derive that

B−1 = INe +AF−1AH , (9.64)

where F ∈ C2×2 is defined in (9.63). Therefore we have simplified the

computation of (CHC)−1 ∈ CNe+2×Ne+2 into simple matrices products and

the inversion of 2 matrices of size 2×2, which can be efficiently computed. We

can now rewrite (9.58) as

wop = C(CHC)−1g =[

U... S

][

B−1 −B−1AD−1

−F−1AH F−1

][

0Ne×1

fk

]

=[

U... S

][

−B−1AD−1fk

F−1fk

]

(9.65)

= −UB−1AD−1fk + SF−1fk. (9.66)

Substituting

D−1 = (SHS)−1 and (9.67)

A = UHS (9.68)

into (9.66), we have that the final simplified solution wop using method 2 is

given bywop = −UB−1UHS(SHS)−1fk + SF−1fk, (9.69)

where

F = SHS− SHUUHS ∈ C2×2, (9.70)

B−1 = INe +UHSF−1SHU ∈ CNe×Ne , (9.71)

U ∈ CM×Ne is the matrix whose columns are the Ne eigenvectors of Q

associated to its Ne largest eigenvalues and S and fk are defined in (9.25)

and (9.29) respectively.

If the communication receiver is moving in the u-space and we have

its displacement information, ∆u, we can compute the new beamforming

DBD



Num. of complex products Num. of complex additionsComputation of the O(M3) O(M3)eigenvectors of Q

SHS︸︷︷︸

D

4M 4(M − 1) = 4M − 4

D−1︸︷︷︸

Din

6 1

UHS︸︷︷︸

A

2NeM 2Ne(M − 1) = 2NeM − 2Ne

D−AHA︸︷︷︸

F

4Ne 4Ne

INe +AF−1AH

︸︷︷︸

B−1

2N2e + 4Ne + 6 N2

e + 3Ne + 1

SF−1fk︸︷︷︸

X

2M + 4 M + 2

UB−1ADinfk︸︷︷︸

Y

N2e + 2Ne +MNe + 4 N2

e +MNe −M + 2

X−Y 0 MTotal O(M3) + (6 + 3Ne)M+ O(M3) + (5 + 3Ne)M+

20 + 10Ne + 3N2e +2 + 9Ne + 2N2

e

Table 9.3: Complex operations of the proposed method 2.

weighting vector wopk (u +∆u) by substituting Su+∆u (9.39) and fk,u+∆ (9.40)

into equations (9.69), (9.70) and (9.71).

Therefore the updated wopk (u+∆u) is given by

wopk (u+∆u) = wo −wop(u+∆u). (9.72)


pute wopt given Ne is detailed in Table 9.3

(b) Eigenvalue Analysis

Method 2 is especially interesting in terms of computational complexity if

Q has only a few significant eigenvalues. In Fig. 9.6 we show the eigenspectrum

of matrix Q for ud = 0.4226, which implies a fidelity region of 50o, and for

ud = 0.1, which implies a fidelity region of 11o, for different array sizes. We

can see that the number of significant eigenvalues are smaller than the total

number of eigenvalues for all array sizes, which is very convenient for the

proposed method.

We can also experimentally notice from Fig. 9.7 that the ratio between

the significant number of eigenvalues holds a similar proportion to the total

number of eigenvalues for all array sizes. In Fig. 9.7 we show that for different

fidelity regions, the proportion of the number of eigenvalues required to

comprise 95% of all eigenvalues for different array lengths is kept almost the

DBD



Eigenvalue Number0 10 30 50 70 90 110 130

10lo

g 10(λ

)

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

Eigenspectra of Q for

ud=0.1 (fidelity region of 11o)

M=10M=50M=90M=130

Eigenvalue Number0 10 30 50 70 90 110 130

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

20

Eigenspectra of Q for

ud=0.4226 (fidelity region of 50o)

M=10M=50M=90M=130

Figure 9.6: Eigenspectrum of Q for different values of M .

Number of array elements0 20 40 60 80 100 120 140 160

Num

ber

of e

igen

valu

es th

at s

um 9

5% o

f the

tota

l

0

10

20

30

40

50

60

70

80

90

100

ud=0.1ud=0.2ud=0.3ud=0.4ud=0.5ud=0.6

Figure 9.7: Number of eigenvalues necessary to comprise 95% of the sum of alleigenvalues vs. M , for several fidelity regions.

same. This fact motivates us to propose an empirical and efficient method for

computing Ne. Based on the fact that the significant eigenvalues of Q can

rapidly sum up 95% of the trace of Q, which is the sum of all eigenvalues, we

can use this as the value of Ne selection criterium. We can be almost sure that

choosing Ne as the number of eigenvalues that sum up 95% of the trace of Q

will lead to a total error bellow the δ threshold. By saying this, after computing

the eigendecomposition, which has computational complexity of O(M3), one

can choose the Ne that comprises 95% of the trace as

Ne∑

n=1

λn > 0.95M∑

n=i

λn. (9.73)

DBD



X

1√2

| · |

Compwiththres-holds

zs∗(τ − t)

r[i] cix(t, u) + n(t)

t = iTPRI + τ

Figure 9.8: Illustration of the non-coherent ASK receiver.

9.4 Proposed Signalling Strategies

In Subsections 9.4.1 and 9.4.2, we briefly describe our proposed sig-

nalling strategies for embedding amplitude and phase modulation. The sig-

nalling strategy for embedding amplitude modulation to the sidelobe of the

radar beampattern (not the method for generating the beampattern itself) is

identical to the strategy described in 8.1. Therefore, we will explain it in 9.4.1

very briefly and we will repeat the bit error expressions derived in 8.1.2.

The signalling strategy for embedding phase modulation to the sidelobe

of the radar beampattern is indeed different and is explained in 9.4.2. We don’t

assume that two or more orthogonal waveforms are transmitted simultaneously,

but we do assume that the radar is coherent or, in other words, we assume

that the initial carrier phase is the same within a coherent processing interval

(CPI). We propose the, commonly used in digital communications, but new

within the DFRC context, differentially encoded PSK (DPSK) modulation [9].

We derive in Subsection 9.4.2 the bit error rate for a generic binary DPSK

(DBPSK) case. This derivation is not found in the literature and is extremely

necessary within the DFRC radar context.

(a) Proposed Signalling Strategy for SLL Modulation

The output, r[i], of the i-th received RF pulse after demodulation,

matched filtering with the waveform s(t) and sampling at the point of max-

imum (see Fig. 9.8) is given by

r[i] =√

Erciejψ + ni. (9.74)

The non-coherent receiver diagram is depicted in Fig. 9.8. We take the

absolute value of r[i], z = |r[i]|, and compare to the thresholds, a decision is

made for symbol k if z is inside the region Zk, delimited by the thresholds,

Tk−1,k ≤ |r| < Tk,k+1.

The probability of wrong decision, P (E), is given by (8.25) and repeated

here for convenience,

DBD



P (E) = 1− 1

K

K−1∑

k=0

[

Q

(µkσn,Tk−1,k

σn

)

− Q

(µkσn,Tk,k+1

σn

)]

, (9.75)

where µk =∣∣√ErCke

jψ∣∣, Q(a, b) is the Marcum Q function, Tk−1,k and Tk,k+1

are the thresholds that bound the region Zk, Zk = [Tk−1,k, Tk,k+1) and σ2n is

the variance of the real and imaginary parts of ni in (9.74).

(b) Proposed Signalling Strategy for Phase Modulation

For the phase modulation, we will take hand of a differentially encoded

PSK (DPSK) modulation [9]. In the DPSK modulation, the information is

encoded into the phase difference between two successive signals and not into

the absolute phase.

In order to implement a DPSK modulation we need a reference phase,

which is transmitted at the beginning of the transmitted message. Therefore,

a sequence of N pulses, m(t), is represented in its low-pass complex equivalent

representation by

m(t) =√

2Etejψ

N∑

i=0

ci(u)s(t− iTPRI), (9.76)

where ci(u) ∈ {Ck(u) = wHk s(u)}Kk=1, Et is the RF pulse energy, ψ is the

initial carrier phase, wk is one of the K beamforming vectors triggered to be

transmitted during the i-th pulse, s(u) is the array steering vector pointed

towards direction u, s(t) is the radar waveform, with bandwidth W << fc,

normalized to unitary energy and TPRI is the radar pulse repetition interval.

Note that, in order to transmit N symbols, it is necessary to transmit

N + 1 absolute phases, i.e. N + 1 pulses. The information phase transmitted

during the i-th pulse isφi = φi − φi−1, (9.77)

where φi and φi−1 are, respectively, the absolute phase of the i-th and i− 1-th

pulses at the receiver. In the absence of noise φi belongs to the set, D, defined

asD = {φk}Ki=1 =

{2π

K(k − 1) + Φ

}K

i=1

, (9.78)

where Φ is an arbitrary constant phase.


beampattern, the output, r[i], after demodulation, matched filtering with the

waveform s(t) and sampling at the point of maximum (see Fig. 9.9) is given

by

r[i] =√

Erejψci(u) + ni, (9.79)

DBD



Decisionbasedonsectors

ℜ{·}

ℑ{·}1√2

r[i]

Delay

s∗(τ − t)

r[i]r∗[i − 1]x(t, u) + n(t)

XX

{·}∗

t = iTPRI + τ

Figure 9.9: Diagram of the equivalent low-pass receiver for DPSK modulatedsignals.

where Er is the received energy which was transmitted isotropically by the

transmitter, considering the propagation attenuation effect, and ni is an

independent complex Gaussian random variable with zero mean and variance

σ2ni

= σ2n = N0.

We can write ci(u) as

ci(u) = |ci(u)|ejαi, (9.80)

thus, substituting (9.80) into (9.79) we have

r[i] =√

Erejψ|ci(u)|ejαi + ni, (9.81)

=√

Er|ci(u)|ejφi + ni. (9.82)

The delayed r[i], r[i− 1] is given by

r[i− 1] =√

Er|ci−1(u)|ejφi−1 + ni−1, (9.83)

where σ2ni−1

= σ2ni

= σ2n = N0 and E[nin

∗i−1] = 0.

Fig. 9.9 shows a diagram of the equivalent low-pass receiver for DPSK

modulated signals.

The receiver bases its decision on the difference between the phases of

the two variables (9.82) and (9.83), that is,

∆βi = arg[z[i]], (9.84)

where

z[i] = r[i]r∗[i− 1], (9.85)

= Er|ci(u)||ci−1(u)|ej(φi−φi−1) +N [i], (9.86)

where N [i] is non-Gaussian complex noise.

DBD



Figure 9.10: Illustration of a D8PSK modulation and its decision regions.

When u = uc we have that

|ci| = |ci−1| and (9.87)

φi − φi−1 ∈ {φk}Kk=1. (9.88)

In these conditions (9.85) resumes to

z[i] = Er|ci|2ej(φi−φi−1) +N [i], (9.89)

= Esejφk +N [i], (9.90)

where Es is the symbol energy at the receiver and φk is a phase from the

desired PSK constellation. Thus, the values assumed by ∆βi in the absence of

noise belong to the PSK constellation.

A correct decision is made if the point represented by ∆βi lies inside

a sector of width 2π/M centered around the correct value of ∆βi, which is

φi ∈ {φk}Ki=1. In Fig. 9.10 it is depicted a D8PSK constellation and the decision

regions described by Zk, k = 1, . . . , 8. A right decision is made for φk if ∆βi

lies inside the sector described by Zk.

Performance of the Non-Coherent DBPSK Receiver When Two ArbitrarySymbols Are Transmitted

We are especially interested in the binary case, DBPSK, where, towards

uc, φi = φi − φi−1 may belong to the set D = {φk}2i=1 = {0, π}. But what

happens if the transmitted pulses have arbitrary amplitudes and arbitrary

phase differences, φk ∈ {0, δ}, δ ∈ [0, 2π)?

Within the DFRC radar context, the classic DBPSK modulation arises,

DBD



ideally, only towards uc, towards the other directions we have arbitrary

amplitudes and phase differences. This arbitrariness happens due to the non

predictable behavior of the beamformers wk, k = 1, . . . , K, in directions other

than the receiver’s direction. Though it is possible to find in the literature BER

expressions for the classic DBPSK case, we haven’t found a closed form for the

DBPSK performance when two arbitrary amplitudes and phase differences are

chosen for representing bits “0” and “1”, possibly because it is not a practical

situation in communications. Therefore, we derive, in this thesis this general

DBPSK performance expression.

This derivation is detailed in the Appendix. A. At this point, we present

only the final result. The error probability, P (e), for the DBPSK modulation

when two arbitrary symbols, C0 and C1, are transmitted is given by

P (e) =1

8e−Er|C0|

2

4σ2w +1

8e−Er |C1|

2

4σ2w

+1

2

∫ +∞

0

QM

( √Er|C1+C0|

2

σw,α

σw

)

α

σ2w

e−α2+

Er |C1−C0|2

42σ2w I0

(

α√Er|C1−C0|

2

σ2w

)

dα,

(9.91)

where σ2w = σ2

n/2 = N0/2 and QM(a, b) is the Marcum Q function.

Note that for the classic DBPSK case, i.e., when C0 = −C1, equation

(9.91) reduces to the P (e) of the classical DBPSK case [9],

P (e) =1

2e−Er|C0|

2

4σ2w =1

2e−EsN0 . (9.92)

DBD


10DFRC Simulations

In this chapter, we show computer simulations of the proposed radar-

embedded sidelobe modulation techniques. We compare the proposed tech-

niques with and without the derivative constraint and we also compare them

with the methods present in the literature described in Chapter 8.

We also present comparisons of the computational complexity in Section

10.7.

10.1 Example of the Proposed Method 1 for Sidelobe AmplitudeModulation

We consider a ULA consisting of M = 10 elements spaced by half

a wavelength. We assume there is a given transmit beampattern and that

the radar primary function takes place at the mainbeam, more precisely,

the interval u ∈ [−0.1736, 0.1736], which corresponds to θ ∈ [80o, 100o].

We also assume that the sidelobes cannot exceed -20 dB and that the

transition intervals are u ∈ [−0.4226,−0.1736] and u ∈ [0.1736, 0.4226],

which correspond to θ ∈ [65o, 80o] and θ ∈ [100o, 115o]. We define the

transition intervals and the mainbeam as the “region of fidelity”, where we

want the new beamforming vectors under design to match the given profile.

The communication receiver is located towards direction uc = −0.6, or

equivalently θc = 126o, and the transmitted symbols belong to the constellation

C = {C1, C2} = {√10−20/10,

√10−40/10}, where

√10−20/10 and

√10−40/10 are

associated to bits “1” and “0” respectively.

Fig. 10.1 shows the beamformers for the described example using the

method of SLL of [5] from the Villanova Center of Advanced Communications

lab explained in 8.1. In Fig. 10.1, w1,v embeds C1 towards uc and w2,v embeds

C2 towards uc.

For the described example, Fig. 10.2 depicts the power pattern of the pro-

posed beamformers, w1, which embeds C1 towards uc, and w2, which embeds

C2 towards uc, according to proposed Method 1. We used the beamformer w1,v

of Fig. 10.1 as the given reference beamformerwo for our proposed method. Fig.

10.2 shows the given transmit output power profile (generated by wo = w1,v)

in green and the two power patterns that correspond to the two communica-

DBD


Chapter 10. DFRC Simulations 156

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

w1,v

w2,v



Figure 10.1: Beamformers that embed a binary amplitude constellation towardsuc = −0.6 using the method of [5].

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.3

wo

w1

w2



Figure 10.2: Power pattern of beamformers w1 and w2 that embed a binaryamplitude constellation towards uc = −0.6 generated using the proposedMethod 1 for α = 0.3.

tion symbols for the proposed Method 1 with first derivative null constraint

and α = 0.3. We chose this value of α because it led to better mainlobe fit

without crossing the permitted sidelobe level.

We can see in Fig. 10.2 that around u = −0.6 the beampatterns generate

a small plateau at their specific communication constraints. We can also notice

that the mainbeam is closely matched by the two beamforming vectors and

none of them lead to sidelobe levels higher than the permitted level. When α

gets closer to 1, the sidelobes cross the permitted level in a few areas.

DBD



α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9e

MB (

%)

1

1.5

2

2.5

3

α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

eT (

%)

4

4.5

5

5.5

6

Figure 10.3: Mainbeam and total rms error vs. α due to beamformers w1(α)and w2(α) depicted in Fig. 10.2 for α = 0.3.

Fig. 10.3 illustrates the behavior of the normalized RMS error of the

mainbeam vs. α and the total normalized rms error vs. α due to beamformers

w1 and w2 computed using Method 1 for different values of α, w1(α) and

w2(α). The normalized rms error at the mainbeam, eMB(α), is

eMB(α) =

√

1

2

(wo −w1(α))HQ(wo −w1(α)) + (wo −w2(α))HQ(wo −w2(α))

wHo Qwo

,

(10.1)

and the total rms normalized error, eT(α), is

eT(α) =

√

1

2

||(wo −w1(α))||2 + ||(wo −w2(α))||2||wo||2

. (10.2)

The behavior of the curves of eMB(α) and eT(α) depicted in Fig. 10.3 is

as expected. As α increases, the total error in the mainbeam decreases, and as

we force the beamformers to fit tightly the mainbeam, the sidelobes get more

erroneous, increasing the total error. These curves give a good rule of thumb

for setting α to achieve a desired tradeoff between sidelobe level and mainbeam

fit.

(a) Recursive Version of Method 1

Fig. 10.4 depicts the power pattern of the proposed beamformers, w1,r

and w2,r, that embed the binary SLL modulation towards uc = −0.6 according

to the recursive version of the proposed Method 1 for 100 iterations. We can

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.3

wo

w1,r

w2,r

fidelity interval


Comm. Direction

Figure 10.4: Power pattern of beamformers w1,r and w2,r that embed a binaryamplitude constellation towards uc = −0.6 generated using the recursiveversion of the proposed Method 1 for α = 0.3.

note that Figs. 10.4 and 10.2 look exactly the same.

Fig. 10.5 depicts the squared error vs. number of iterations, n. The

squared errors, e1(n) and e2(n), due to the convergence of w1,r(n) and w2,r(n)

respectively are computed as

e1(n) = ||w1 −w1,r(n)||2, (10.3)

e2(n) = ||w2 −w2,r(n)||2. (10.4)

DBD



Number of iterations0 10 20 30 40 50 60 70 80 90 100

Squ

ared

err

or

×10-4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

squared error due to w1,r

squared error due to w2,r

Figure 10.5: Squared error vs. number of iterations of the beamformers w1,r

and w2,r, depicted in Fig. 10.4, generated using the recursive version of theproposed Method 1 for α = 0.3.

(b) Method 1 without the Derivative Constraint

Fig. 10.6 depicts the same example of Fig. 10.2 for the case without the

first derivative null constraint and α = 0.5. We chose this value of α because, for

this case, it led to better mainlobe fit without crossing the permitted sidelobe

level. The beamformers generated using the proposed Method 1 without the

first derivative null constraint are denoted by w1 and w2. We can notice

that the mainbeam is matched exactly by the two beamforming vectors and

none of them lead to sidelobe levels higher than the permitted level. We can

also note that our simple, closed form solution generates practically the same

beampatterns as the method of [5], depicted in Fig. 10.1, which uses interior

point technique for computing the solution to their problem formulation.

Fig. 10.7 illustrates the behavior of the normalized rms error of the

mainbeam vs. α and the total normalized rms error vs. α for the case without

the first derivative null constraint due to beamformers w1 and w2. We can note

that the behavior is the same as for the case with the derivative null constraint

depicted in Fig. 10.3.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.5

wo

w1

w2

Comm. Direction

fidelity interval


Figure 10.6: Beamformers that embed a binary amplitude constellation towardsuc = −0.6 using the proposed Method 1 for α = 0.5 without the firstderivative null constraint.

α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

eM

B (

%)

0.2

0.4

0.6

0.8

α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

eT (

%)

1.9

1.95

2

2.05

2.1

Figure 10.7: Mainbeam and total rms error vs. α for the case without thefirst derivative null constraint due to beamformers w1(α) and w2(α) depictedin Fig. 10.6.

(c) Recursive Version of Method 1 without the Derivative Constraint


and w2,r, that embed the binary SLL modulation towards uc = −0.6 according

to the recursive version of the proposed Method 1 for 100 iterations. We can

note that Figs. 10.8 and 10.6 look exactly the same.

Fig. 10.9 depicts the squared error, e1(n) and e2(n), due to the conver-

gence of w1,r(n) and w2,r(n) respectively, vs. number of iterations.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.5

wo

w1,r

w2,r

Comm. Direction

fidelity interval


Figure 10.8: Power pattern of beamformers w1,r and w2,r that embed a binaryamplitude constellation towards uc = −0.6 generated using the recursiveversion of the proposed Method 1 for α = 0.5 for the case without the firstderivative null constraint.

0 20 40 60

×10-30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Squared error due to w1,r

0 20 40 60

×10-5

0

0.5

1

1.5

2

2.5

3



and w2,r, depicted in Fig. 10.8, generated using the recursive version of theproposed Method 1 for α = 0.5 for the case without the first derivative nullconstraint.

10.2 Example of the Proposed Method 2 for Sidelobe AmplitudeModulation

In this section we repeat the data of the example of Section 10.1 and

we apply Method 2, described in Section 9.3, for generating the beamformers

that embed the amplitude modulation. The power patterns of the beamformers

generated using Method 2, w1,e and w2,e, are depicted in Fig. 10.10 for Ne = 5

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0N

e = 5

wo

w1,e

w2,e

Comm. Direction

fidelity interval


Figure 10.10: Power pattern of beamformers w1,e and w2,e that embed a binaryamplitude constellation towards uc = −0.6 generated using the proposedMethod 2 for Ne = 5.

(which is the value that sum up 95% of the trace of Q).


mainbeam eMB(Ne) vs.Ne and the total normalized rms error eT(Ne) vs.Ne due

to beamformers w1,e and w2,e, computed using Method 2 for different values

of Ne, w1,e(Ne) and w2,e(Ne). The mainbeam error, eMB(Ne), is computed as

eMB(Ne) =√

1

2

(wo −w1,e(Ne))HQ(wo −w1,e(Ne)) + (wo −w2,e(Ne))HQ(wo −w2,e(Ne))

wHo Qwo

,

(10.5)

and the total rms normalized error, eT(Ne), is computed as

eT(Ne) =

√

1

2

||(wo −w1,e(Ne))||2 + ||(wo −w2,e(Ne))||2||wo||2

. (10.6)

The behavior of the curves of eMB and eT seen in Fig. 10.11 is as expected.

We can compare Ne of Method 2 to α of Method 1. Similarly to α, as Ne

increases, the total error in the mainbeam decreases, and as we force the

beamformers to fit tightly the mainbeam, the sidelobes get more erroneous,

increasing the total error.

Comparing the power patterns generated using Methods 1 and 2 in Figs.

10.2 and 10.10 respectively, we can note that they differ a little visually. The

mainbeam and total errors for Method 1 and α = 0.3 are of eMB(α = 0.3) =

DBD



Ne

1 2 3 4 5 6 7 8e

MB (

%)

0

1

2

3

Ne

1 2 3 4 5 6 7 8

eT (

%)

0

50

100

150

200

Figure 10.11: Mainbeam and total rms error vs. Ne due to beamformersw1,e(Ne) and w2,e(Ne) depicted in Fig. 10.10.

2.08% and eT (α = 0.3) = 4.36% respectively, while for Method 2 with Ne = 5

they are of eMB(Ne = 5) = 1.63% and eT (Ne = 5) = 5.39% respectively.

The difference between the two methods is very small. Anyway, as long as the

sidelobe levels are below the maximum permitted level and the mainbeam is

fit properly by the designed beamformers, both methods are useful for DFRC

radars.

(a) Method 2 without the Derivative Constraint

Fig. 10.12 depicts the same example of Fig. 10.10 for the case without

the first derivative null constraint and Ne = 5. The value of Ne does not

change because the number of eigenvalues necessary to reduce the error at the

mainbeam depends only on matrix Q, which in its turn depends only on the

array geometry and the sector of the mainbeam. The beamformers generated

using the proposed Method 2 without the first derivative null constraint are

denoted here by w1,e and w2,e. We can notice that the mainbeam is matched

exactly by the two beamforming vectors and none of them lead to sidelobe

levels higher than the permitted level.


mainbeam vs. Ne and the total normalized rms error vs. Ne for the case

without the first derivative null constraint due to beamformers w1,e and w2,e.

Comparing the power patterns generated using Methods 1 and 2 in Figs.

10.6 and 10.12 respectively, for the case without the first derivative null

constraint, we can note that they differ minimally visually. The mainbeam

and total errors for Method 1 and α = 0.5 are of eMB(α = 0.5) = 0.38% and

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0N

e = 5

wo

w1,e

w2,e

Comm. Direction

fidelity interval


Figure 10.12: Beamformers, w1,e and w2,e, that embed a binary amplitudeconstellation towards uc = −0.6 using the proposed Method 2 for Ne = 5without the first derivative null constraint.

Ne

1 2 3 4 5 6 7 8

eM

B (

%)

0

0.5

1

Ne

1 2 3 4 5 6 7 8

eT (

%)

0

5

10

Figure 10.13: Mainbeam and total rms error vs. Ne for the case without

the first derivative null constraint due to beamformers w1,e(Ne) and w2,e(Ne)depicted in Fig. 10.12.

eT (α = 0.5) = 1.95% respectively, while for the Method 2 with Ne = 5 they are

of eMB(Ne = 5) = 0.49% and eT (Ne = 5) = 1.94% respectively. The difference

between the two methods is indeed very small.

The proposed Method 2 without the first derivative null constraint is

also able to generate practically the same beamformers as the ones achieved

using the complex method of of [5], depicted in Fig. 10.1, in a much simpler

manner.

DBD



Z0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

prob

abili

ty d

ensi

ty fu

nctio

n

0

5

10

15

20

25

30

35

40

45

pz|1

(Z)

pz|0

(Z)

bit 1bit 0

Figure 10.14: Conditional probability density function of the symbols trans-mitted using the binary constellation C = {C1, C2} = {

√10−20/10,

√10−40/10}

and SNR = 15 dB.

10.3 Robustness Analysis for the Proposed Sidelobe AmplitudeModulation Methods

Fig. 10.14 depicts the conditional density function of the received symbols

for the embedded constellation C = {C1, C2} = {√10−20/10,

√10−40/10},

where C1 =√10−20/10 is associated to bit “1” and symbol C2 =

√10−40/10

corresponds to bit “0”, for a SNR of 15 dB.

The SNR in dB is defined as

SNR = 10 log100.5|C1|2 + 0.5|C2|2

σ2n

, (10.7)

where σ2n is the noise variance of the received samples (after demodulation and

matched filtering).

From Fig. 10.14 we can check that the threshold for computing the

analytical angular BER for a SNR of 15 dB is T = 0.056, which is the point

where the conditional density functions cross each other. Having defined the

decision threshold, we can now compute the angular BER for all beamformers

seen so far using the analytical BER expression of (9.75).

Fig. 10.15 depicts the angular BER (more precisely the BER in u-

space plane) of the proposed Method 1, with and without the derivative null

constraint (α = 0.3 and α = 0.5 respectively), and the method of the Villanova

Center [5]. As the results of Method 1 and its recursive version are the same,

assuming that convergence is achieved, we will depict only the results for

Method 1 in its closed form. Fig. 10.16 depicts the angular BER of the proposed

Method 2, with and without the derivative null constraint (Ne = 5 for both

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

BE

R

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Method 1Method 1 without the derivative null ocnstraintMethod of Villanova Center

Rx. Direction

Figure 10.15: BER × u, for embedded amplitude modulation at uc = −0.6,for the proposed Method 1 with and without the derivative constraint and themethod of [5].

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

BE

R

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Method 2Method 2 without the derivative null constraintMethod of Villanova Center

Rx. Direction

Figure 10.16: BER × u, for embedded amplitude modulation at uc = −0.6,for the proposed Method 2 with and without the derivative constraint and themethod of [5].

cases), and the method of [5].

We can confirm by looking at Figs. 10.15 and 10.16 that the results of the

proposed methods without the derivative null constraint and the method of [5]

are very similar, as expected, given that the beampatterns of both methods are

practically the same. The angular BER curve falls rapidly with a very narrow

pit to the operational BER at the direction of the communication receiver.

On the other hand, the outcome of the proposed methods with first

DBD



u-0.7 -0.68 -0.66 -0.64 -0.62 -0.6 -0.58 -0.56 -0.54 -0.52 -0.5

BE

R

10-8

10-6

10-4

10-2

100Method 1Method 1 without the derivative null ocnstraintMethod of Villanova Center

10 minutes ∆u

20 minutes ∆u

Figure 10.17: Zoom of the BER × u, for embedded amplitude modulationat uc = −0.6, for the proposed Method 1 with and without the derivativeconstraint and the method of [5].

derivative null constraint is a flat operational BER over a small region. This

property makes the proposed techniques robust to small deviation errors on

the preassigned direction, which is fundamental for the communication system

service, since involuntary small deviation errors in the relative positioning of

the communication receiver and the radar is likely to happen.

Fig. 10.17 assumes the data of the sidelobe amplitude example of Section

9.1. The communication receiver is in a corvette moving away from the

radar mainbeam with a cruising speed of 17 knots or equivalently 8.75 m/s

in a direction tangential to the circumference of radius 180 Km centered

at the radar. In 10 minutes the relative angle between the radar and the

communication receiver would have increased of ∆u = −0.025. Checking the

angular BER in Fig. 10.15 we can see that the BER of the method of the

Villanova Center would go from 3.17× 10−7 to 1.1× 10−3 in only 10 minutes.

In 20 minutes ∆u = −0.05 and the BER goes to 0.27. The BER of the proposed

method, on the other side, goes from 3.17× 10−7 to 4.35× 10−7 in 10 minutes

and to 5.24× 10−6 in 20 minutes, which keeps the communication operational

during the whole time.

This BER dynamic is illustrated in Fig. 10.17, a zoom of Fig. 10.15.

10.4 Example of the Proposed Method 1 for Sidelobe Phase Modulation

In this example, we will embed the BPSK constellation C = {φ1, φ2} =

{0o, 180o} towards uc = −0.6. We will use the same reference beamformer, wo,

as the one used for the amplitude modulation example of the former Section

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

w0

w2

w1

Comm. Direction


fidelity interval

Figure 10.18: Beamformers that embed a DBPSK constellation towards uc =−0.6 using the method of [4].

10.1 (wo is beamformer w1,v of Fig. 10.1).

In Fig. 10.18 we simulate the method of common reference from the

Villanova Center, [4], explained in 8.2.2, using w1,v of Fig. 10.1 as the common

reference beamformer, w0. The common reference, w0, is always used with

its own sensor array and is transmitted in tandem with a certain waveform.

Beamformers w1 and w2 are designed according to the optimization problem

(8.43) and are applied to another sensor array and transmitted in tandem

with an orthogonal waveform according to the current message. Beamformer

w1 embeds φ1 = 0o into the phase difference between the beampattern of

w1 and w0 towards uc and w2 embeds φ1 = 180o into the phase difference

between the beampattern of w2 and w0 towards uc. We won’t simulate the

method of rotational invariance because this method doesn’t lead to secure

communications as discussed in 8.2.1.

We can see from Fig. 10.18 that the sidelobes are just in the limit of

crossing the permitted level line. That is because the method of [4], differently

from their proposed method for sidelobe amplitude modulation, does’t impose

this sidelobe constraint to the optimization problem formulation. Since they

use interior point technique for solving their optimization problem, the reason

for not adding this important constraint is not clear to us.

Fig. 10.19 shows the phase difference of the beamformer pairs of Fig.

10.18 (w1,w0) and (w2,w0) in the u-space. We can note that, towards uc =

−0.6, the phase difference between w1 and w0 is zero, which corresponds to

embedding φ1 = 0, and the phase difference between w2 and w0 is 180o, which

corresponds to embedding φ2 = 180o.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pha

se in

deg

rees

-150

-100

-50

0

50

100

150phase difference due to beamformers pair (w

1, w

0)

phase difference due to beamformers pair (w2, w

0)

communication receiver direction

Figure 10.19: Phase difference of the beamformer pairs of Fig. 10.18 (w1,w0)and (w2,w0) in the u-space.

We propose a DBPSK signaling strategy that differs from the strategies

for embedding sidelobe phase modulation existent in the DFRC literature

so far. The DBPSK modulation is explained in 9.4.2, it can be summarized

as: when bit “0” is triggered for transmission, the same beamformer as the

last pulse is used and when bit “1” is triggered for transmission the other

beamformer, different from the one used for the last pulse, is chosen. For

the described example, Fig. 10.20 depicts the power pattern of the proposed

beamformers, w1 and w2, that embed the DBPSK modulation towards uc =

−0.6 according to proposed Method 1.

Fig. 10.20 shows the given transmit output power profile in green and

the two power patterns that correspond to the two beamformers generated

using the proposed Method 1 with first derivative null constraint and α = 0.6.

We chose this value of α because it led to better mainlobe fit without crossing

the permitted sidelobe level. The beampattern generated by w2 has a phase

shift towards uc of 180o in relation to the beampattern generated by w1 and

vice versa. Both w1 and w2 try to match as close as possible the beampattern

of the reference beamformer, wo, with a tighter adjustment at the mainlobe.

We can notice that the mainbeam is closely matched by the two beamforming

vectors and none of them lead to sidelobe levels higher than the permitted

level. When α gets closer to 1, the sidelobes cross the permitted level in a few

areas.

Fig. 10.21 shows the phase difference of the beamformer pair of Fig.

10.20 (w2,w1) in the u-space. We can note that towards uc = −0.6 the phase

difference between w2 and w1 is, as expected, of 180o.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.6

wo

w1

w2

fidelity interval


Comm. Direction

Figure 10.20: Power pattern of beamformers w1 and w2 that embed a DBPSKmodulation towards uc = −0.6 generated using the proposed Method 1 forα = 0.6.

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pha

se in

deg

rees

-200

-150

-100

-50

0

50

100

150

200Phase difference due to beamforers w

1 and w

2

Rx. Direction

Figure 10.21: Phase difference of the beamformer pair of Fig. 10.20 (w2,w1) inthe u-space.

We can note from observing Figs. 10.20 and 10.21 that around u = −0.6

the beampatterns generate a small plateau at their specific communication

constraints which keeps the phase difference stable within this plateau.

Fig. 10.22 depicts the behavior of the normalized rms error of the

mainbeam eMB(α) vs. α and the total normalized rms error eT(α) vs. α due to

beamformers w1 and w2. The behavior of the curves of eMB(α) and eT(α) seen

in Fig. 10.22 is as expected. As α increases, the total error in the mainbeam

decreases, and as we force the beamformer to fit tightly the mainbeam, the

DBD



α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9e

MB (

%)

1

1.5

2

2.5

3

α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

eT (

%)

5.5

6

6.5

7

7.5

Figure 10.22: Mainbeam and total rms error vs. α due to beamformers w1(α)and w2(α) depicted in Fig. 10.20.

SNR (dB)0 5 10 15

BE

R

10-15

10-10

10-5

100

Analytical BER for the DBPSK systemSimulated BER for the DBPSK systemSimulated BER for the system of commom reference

Figure 10.23: Simulated and analytical BER for a DBPSK system.

sidelobes get more erroneous, increasing the total error.

Fig. 10.23 depicts the BER for the proposed binary DPSK system, both

analytical and simulated, and the simulated BER for the method of commom

reference from the Villanova Center [4]. We can note from Fig. 10.23 that the

simulated and analytical curves are in agreement and we can also note that,

though the method of commom reference is more sophisticated, it achieves the

same BER as the simple DBPSK signalling strategy proposed here for sidelobe

phase modulation.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.6

wo

w1,r

w2,r

Comm. Direction

fidelity interval


Figure 10.24: Power pattern of beamformers w1,r and w2,r that embed a binaryDPSK constellation towards uc = −0.6 generated using the recursive versionof the proposed Method 1 for α = 0.6.

0 10 20 30 40 50 60 70 80 90 100

×10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Squared error due to w1

Squared error due to w2


and w2,r, depicted in Fig. 10.24, generated using the recursive version of theproposed Method 1 for α = 0.6.

(a) Recursive Version of Method 1


and w2,r, that embed the binary DPSK modulation towards uc = −0.6

according to the recursive version of the proposed Method 1 for 100 iterations.

We can note that Figs. 10.24 and 10.20 look exactly the same. Fig. 10.25

depicts the squared error vs. number of iterations.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.1

wo

w1

w2Comm. Direction

fidelity interval


Figure 10.26: Beamformers that embed a binary DPSK constellation towardsuc = −0.6 using the proposed Method 1 for α = 0.1 without the firstderivative null constraint.

(b) Method 1 without the Derivative Constraint


the first derivative null constraint and α = 0.1. We can notice that the

mainbeam is matched exactly by the two beamforming vectors, w2 and w1.

We had to use a very small α so that none of beampatterns lead to sidelobe

levels higher than the permitted level. Even though, the sidelobes are just in

the limit, like the beampatterns generated by the method of common reference

from the Villanova Center, [4], depicted in Fig. 10.18.


10.26 (w2,w1) in the u-space. We can note that towards uc = −0.6 the phase

difference between w2 and w1 is, as expected, of 180o.


mainbeam vs. α and the total normalized rms error vs. α for the case without

the first derivative null constraint due to beamformers w1 and w2.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pha

se in

deg

rees

-200

-150

-100

-50

0

50

100

150

200

Phase difference due to beamforers w1 and w2

Rx. Direction


α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

eM

B (

%)

0.5

1

1.5

α

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

eT (

%)

4.2

4.4

4.6

4.8

Figure 10.28: Mainbeam and total rms error vs. α for the case without thefirst derivative null constraint due to beamformers w1(α) and w2(α) depictedin Fig. 10.26.

(c) Recursive Version of Method 1 without the Derivative Constraint


and w2,r, that embed the binary DPSK modulation towards uc = −0.6

according to the recursive version of the proposed Method 1 for 100 iterations.

We can note that Figs. 10.29 and 10.26 look exactly the same.

Fig. 10.30 depicts the squared error, e1 and e2, due to the convergence

of w1,r and w2,r respectively, vs. number of iterations.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.1

wo

w1,r

w2,r

Comm. Direction

fidelity interval


Figure 10.29: Power pattern of beamformers w1,r and w2,r that embed a binaryDPSK constellation towards uc = −0.6 generated using the recursive versionof the proposed method 1 for α = 0.1 for the case without the first derivativenull constraint.

0 20 40 60

×10-5

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


0 20 40 60

×10-31

0

0.2

0.4

0.6

0.8

1

1.2

1.4



and w2,r, depicted in Fig. 10.29, generated using the recursive version of theproposed Method 1 for α = 0.1 for the case without the first derivative nullconstraint.

10.5 Example of the Proposed Method 2 for Sidelobe Phase Modulation

In this section we repeat the data of the example of Section 10.4 and we

apply the Method 2, described in Section 9.3, for generating the beamformers

that embed the DBPSK modulation. The power patterns of the beamformers

generated using Method 2, w1,e and w2,e, are depicted in Fig. 10.31 for Ne = 5,

which is the number of eigenvalues that sum up 95% of the trace of Q. Note

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Ne = 5

wo

w1,e

w2,e

Comm. Direction

fidelity interval


Figure 10.31: Power pattern of beamformers w1,e and w2,e that embed a binaryDPSK constellation towards uc = −0.6 generated using the proposed Method2 for Ne = 5.

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pha

se in

deg

rees

-200

-150

-100

-50

0

50

100

150

200Ne = 5

Phase difference due to beamforers w1,e

and w2,e

Rx. Direction

Figure 10.32: Phase difference of the beamformer pair of Fig. 10.31 (w2,e,w1,e)in the u-space.

that as long as the mainbeam sector and the array remain the same, the number

of eigenvalues that sum up 95% of the trace of Q is the same.


10.31 (w2,e,w1,e) in the u-space. We can note that towards uc = −0.6 the

phase difference between w2 and w1 is, as expected, of 180o.

We can note from observing Figs. 10.31 and 10.32 that around u = −0.6

the beampatterns generate a small plateau at their specific communication

constraints which keeps the phase difference stable within this plateau.

DBD



Ne

1 2 3 4 5 6 7 8e

MB (

%)

0

1

2

3

4

Ne

1 2 3 4 5 6 7 8

eT (

%)

0

100

200

300

Figure 10.33: Mainbeam and total rms error vs. Ne due to beamformersw1,e(Ne) and w2,e(Ne) depicted in Fig. 10.31.


mainbeam vs. Ne and the total normalized rms error vs.Ne due to beamformers

w1,e and w2,e.

The behavior of the curves of eMB(Ne) and eT(Ne) seen in Fig. 10.33 is

as expected. We can compare Ne of Method 2 to α of Method 1. Similarly to

α, as Ne increases, the total error in the mainbeam decreases, and as we force

the beamformer to fit tightly the mainbeam, the sidelobes get more erroneous,

increasing the total error.

Comparing the power patterns generated using Methods 1 and 2 in

Figs. 10.20 and 10.31 respectively, we can note that they are very similar.

The mainbeam and total errors for Method 1 and α = 0.6 are of eMB(α =

0.6) = 1.86% and eT (α = 0.6) = 6.09% respectively, while for the Method

2 with Ne = 5 they are of eMB(Ne = 5) = 1.88% and eT (Ne = 5) = 6.63%

respectively. The difference between the two methods is very small.

(a) Method 2 without the Derivative Constraint


the first derivative null constraint and Ne = 5. The beamformers generated

using the proposed Method 2 without the first derivative null constraint

are denoted here by w1,e and w2,e. We can notice that the original power

pattern is matched exactly by the beamforming vector w1,e, but beamforming

weighting vector w2,e generated a small hump around u = −0.5 over the

maximum permitted sidelobe level. Therefore we reduced the number of null

eigenvectors to 3, Ne = 3, in order to reduce this hump. The power patterns

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0N

e = 5

wo

w1,e

w2,e

Comm. Direction


fidelity interval

Figure 10.34: Beamformers, w1,e and w2,e, that embed a binary DPSK constel-lation towards uc = −0.6 using the proposed Method 2 for Ne = 5 without

the first derivative null constraint.

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0N

e = 3

wo

w1,e

w2,e

Comm. Direction

fidelity interval

Figure 10.35: Beamformers, w1,e and w2,e, that embed a binary DPSK constel-lation towards uc = −0.6 using the proposed Method 2 for Ne = 3 without

the first derivative null constraint.

using Ne = 3 are depicted in Fig. 10.35. Using Ne = 3, the beampatterns

generated using Method 2 without the derivative constraint are very similar to

the beampatterns obtained using Method 1 without the derivative constraint

and α = 0.1 and the beampatterns from the Villanova Center, [4], depicted in

Fig. 10.18.


10.35 (w2,e,w1,e) in the u-space. We can note that towards uc = −0.6 the

phase difference between w2,e and w1,e is, as expected, of 180o.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pha

se in

deg

rees

-200

-150

-100

-50

0

50

100

150

200Ne = 3

Phase difference due to beamforers w1,e and w2,e

Rx. Direction


Ne

1 2 3 4 5 6 7 8

eM

B (

%)

0

0.5

1

1.5

2

Ne

1 2 3 4 5 6 7 8

eT (

%)

0

5

10

15

20

Figure 10.37: Mainbeam and total rms error vs. Ne for the case without

the first derivative null constraint due to beamformers w1,e(Ne) and w2,e(Ne)depicted in Fig. 10.35.


mainbeam vs. Ne and the total normalized rms error vs. Ne for the case

without the first derivative null constraint due to beamformers w1,e and w2,e.

Comparing the power patterns for the case without the first derivative

null constraint generated using Methods 1 and 2 in Figs. 10.26 and 10.35

respectively, we can note that they differ minimally visually. The mainbeam

and total errors for Method 1 and α = 0.1 are of eMB(α = 0.1) = 1.44% and

eT (α = 0.1) = 4.25% respectively, while for the Method 2 with Ne = 3 they are

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

BE

R

10-2

10-1

100Angular BER for SNR = 5 dB

Simulated dataDerived analytical expression

Figure 10.38: Analytical and simulated BER × u, for embedded DBPSKmodulation at uc = −0.6, for the proposed Method 1 without the derivativeconstraint for SNR = 5 dB.

of eMB(Ne = 3) = 1.82% and eT (Ne = 3) = 4.22% respectively. The difference

between the two methods is indeed very small.

10.6 Robustness Analysis for the Proposed Sidelobe Phase ModulationMethods

In this section we will use the derived analytical expression for the

DBPSK modulation when two arbitrary symbols are transmitted. We present

Fig. 10.38 in order to attest that the derived expression in equation (9.91) is

in agreement with the simulated data. Fig. 10.38 depicts the simulated and

the analytical angular BER for the proposed DBPSK signaling strategy using

the beamformers w1 and w2 of Fig. 10.26 for a SNR of 5 dB. We can note

from Fig. 10.38 that the analytical expression is indeed in agreement with the

simulated data, therefore we will use only the analytical expression for our

proposed methods from now on.

Fig. 10.39 depicts the angular BER (more precisely the BER in u-

space plane) of the proposed Method 1, with and without the derivative null

constraint (α = 0.6 and α = 0.1 respectively), and the method of [4]. As

the results of Method 1 and its recursive version are the same, assuming that

convergence is achieved, we will depict only the results for Method 1 in its

closed form. Fig. 10.40 depicts the angular BER of the proposed Method 2, with

and without the derivative null constraint (Ne = 5 and Ne = 3 respectively),

and the method of [4].

We can confirm by looking at Figs. 10.39 and 10.40 that the results of

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

BE

R

10-5

10-4

10-3

10-2

10-1


Method 1Method 1 without the derivative null constraintMethod of common reference from the Villanova Center

Figure 10.39: BER × u, for embedded DBPSK modulation at uc = −0.6, forthe proposed Method 1 with and without the derivative constraint and themethod of [4] for SNR = 10 dB.

u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

BE

R

10-5

10-4

10-3

10-2

10-1



Figure 10.40: BER × u, for embedded DBPSK modulation at uc = −0.6, forthe proposed Method 2 with and without the derivative constraint and themethod of [4] for SNR = 10 dB.

the proposed methods without the derivative null constraint and the method

of [4] are very similar. The angular BER curve falls rapidly with a very

narrow pit to the operational BER at the direction of the communication

receiver. On the other hand, the outcome of the proposed methods with first

derivative null constraint is a flat operational BER over a small region. This

property makes the proposed techniques robust to small deviation errors on

the preassigned direction, which is fundamental for the communication system

DBD



u-0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 -0.45 -0.4

BE

R

10-5

10-4

10-3

10-2

10-1



20 min ∆u

Figure 10.41: Zoom of the BER × u, for embedded DBPSK modulationat uc = −0.6, for the proposed Method 1 with and without the derivativeconstraint and the method of [4] for SNR = 10 dB.

service, since unintentional small deviation errors in the relative positioning of

the communication receiver and the radar is likely to happen.

Fig. 10.41 assumes the same data as the example in 10.3. Checking the

angular BER in Fig. 10.39 we can see that the BER of the method of the

Villanova Center would go from 2.5 × 10−5 to 0.01 in only 20 minutes. The

BER of the proposed method, on the other side, goes from 2.5 × 10−5 to

2.31× 10−5 in 20 minutes, which is a negligible performance degradation.

This BER dynamic is illustrated in a zoom of Fig. 10.39, Fig. 10.41.

10.7 Computational Complexity

Fig. 10.42 depicts a comparison of the total number of complex products

and additions for the proposed Method 1, the recursive version of Method 1

and Method 2, given α and given Ne, according to Tables 9.1, 9.2 and 9.3

respectively.

The procedure of choosing Ne described in (9.73) has computational

complexity of Ne−1+M−1 complex additions and 1 complex product. On the

other hand, in order to determine a good choice of α, one should discretise the α

domain, which is the interval [0,1) into D samples and compute the total error

for all the D possible beamforming filters. This procedure has computational

complexity in the order of O(DM3). Though it seems high, there are efficient

manners of computing a value of α which meets the error at the mainbeam

requirement. Beside this, we have noticed that the error at the mainbeam is

not the only important factor for choosing neither Ne nor α. It is important

DBD



M50 100 150

Num

ber

of C

ompl

ex P

rodu

cts

103

104

105

106

107

M50 100 150

Num

ber

of C

ompl

ex A

dditi

ons

103

104

105

106

107

Method 1Rec. vers. of Met. 1, N

it=10

Rec. vers. of Met. 1, Nit=50

Method 2, ud = 0.1

Method 2, ud = 0.6

Figure 10.42: Comparison of the number of complex products and additionsnecessary for Method 1, recursive version of Method 1 and Method 2, given αand Ne.

to attest if the sidelobes are bellow the permitted level. Therefore, we won’t

consider the preprocessing step of choosing α nor Ne.

For Method 1 and its recursive version, the value of α doesn’t change

the number of operations, once it is chosen. On the other hand, the number

of operations is dependent on Ne for Method 2. Therefore, for Method 2, we

consider two different fidelity sectors, a wide one, of ud = 0.6 and a narrow one

of ud = 0.1. We present two different fidelity sectors because the number of

eigenvalues that sum up 95% of the trace of Q are different. Therefore, these

two cases illustrate extreme cases. We used Ne as the number of eigenvalues

that sum up 95% of the trace of Q for all different array sizes for both fidelity

sectors. The values ofNe can be assessed from Fig. 9.7. For the recursive version

of Method 1 we have considered two different number of iterations, Nit = 10

and Nit = 50, that is because the observed initial squared error is already very

small. A metric based on the radar performance would be more appropriate

for setting this value though.

We can see from Fig. 10.42 that the computational complexity of Methods

1 and 2, for both extreme cases of fidelity sector, given α and Ne, respectively,

are very similar. The number of complex operations of the recursive version of

Method 1 may be lower or higher depending on the number of iterations.

10.8 Clutter Analysis

A controversial topic within sidelobe modulation is the effect of sidelobe

modulation in clutter mitigation techniques. It is said that sidelobe modulation

DBD



would cause Doppler spread, interfering in clutter mitigation methods. In this

section we use the clutter expression normally used in clutter simulations

derived in Section 3.2 and we extend it to the case where sidelobe modulation

is used. Then we examine one clutter mitigation technique, namely, the space-

time MVDR filter, and present simulation results for a few examples.

As explained in Section 3.2, the continuous field of clutter can be

approximated by the superposition of a large number, Nc, of independent

clutter sources that are uniformly distributed in azimuth about the radar.

The location of the l, p-th clutter patch is described by its azimuth, φp,

and range, Rl, (or elevation, θl). The corresponding spatial frequency is

ϑl,p =dλcos(θl) sin(φp) [1].

The clutter amplitudes, αl,p, satisfy E [|αl,p|2] = σ2nξl,p, where σ

2n is the

noise power per element and ξl,p is the clutter to noise ratio (CNR). The

CNR due to the l, p-th clutter patch can be obtained directly from the radar

equation:

ξl,p =GrGtλ

2c |B(ϑl,p)|2Ptσl,p

(4π)3σ2nLsR

4l

, (10.8)

where Gt is the transmitter gain, Gr is the receiver gain, λc is the carrier

frequency, Pt is the isotropically irradiated transmitted power, σl,p is the (l, p)-

th clutter patch cross section, Rl is the range of the (l, p)-th clutter patch, Ls

is the system loss and Pl,p is the power transmitted towards the l, p-th clutter

patch which is dependent on its spatial frequency and B(ϑl,p) is the normalized

beampattern towards ϑl,p,

B(ϑl,p) =B(ϑl,p)

B(ϑmax), (10.9)

where B(ϑmax) is the beampattern towards the direction of maximum.

In the case of SLL modulation, the transmit beamformer vector is

chosen from a set of K possible beamformers at each pulse according to

the communication symbols to be transmitted, which have uniform a priori

probability distribution. It means that the amplitude αl,p is a discrete random

variable. Thus, for the case of sidelobe modulation the clutter to noise ratio

relative to the (l, p)-th clutter patch is given by

ξl,p =GrGtλ

2cPtσl,p

K(4π)3σ2nLsR

4l

K∑

k=1

|Bk(ϑl,p)|2. (10.10)

Assuming that returns from different clutter patches are uncorrelated,

E{αl,pα

∗i,j

}= σ2

nξl,pδl−iδp−j, (10.11)

and assuming unambiguous range, i.e. only range Rl is contributing to the

clutter, the clutter space-time covariance matrix defined in (3.51) is given by

DBD



[1]

Rc = E[c[l]cH [l]

]= σ2

n

Nc∑

p=1

ξl,pv(ϑl,p, fl,p)vH(ϑl,p, fl,p), (10.12)

where v(ϑl,p, fl,p) is the space-time steering vector of the (l, p)-th clutter patch

defined in (3.50).

We can note from (10.10) and (10.12) that the only difference in the

clutter covariance matrix, Rc, when using sidelobe modulation, is that the

clutter to noise ratio is, in fact, computed using the mean value of the various

transmit beampatterns used. Therefore, if the clutter covariance matrix can be

well estimated, we don’t expect to have any degradation in clutter mitigation

techniques that use the inverse of the covariance matrix, like the space-time

MVDR method for example.

In order to illustrate this theoretical result, we will show some simulation

results. In this analysis we consider that the radar platform is not moving

relative to the clutter, va = 0 m/s, i.e, the clutter ridge, β is zero, so that the

clutter is spread only in spatial frequency and not in the Doppler domain. We

use this assumption because as the platform moves, the angular position of

the communication receiver also varies. So, in order to assess realistic results,

one should have to model the variation of the relative communication receiver

direction with respect to the platform movement and change the beamformers

consequently. This is dependent on the type of radar, as airborne radars are

different from landbased radars for example.

In our simulations, we have used the beamformers of Fig. 10.2, repeated

here in Fig. 10.43 with the addition of the mean beampattern. We adapted

the coordinate system of the spatial frequency, ϑ, defined in Fig. 3.2 to be

in agreement with the coordinate system adopted for the generation of the

beamformers, therefore

ϑl,p =d

λccos(θl) cos(φp), (10.13)

where the array geometry and φ are as depicted in Fig. 2.4.

We simulated a radar withK = 10 elements displaced in a ULA separated

by half of the carrier wavelength, which transmits J = 18 pulses per CPI

with a pulse repetition frequency (PRF) of 300 Hz and operating frequency of

fc = 450 MHz. We modeled a ring of ground clutter distant 130 km from the

radar using Nc = 360 clutter patches uniformly distributed in azimuth with a

clutter cross section of σl,p = 2.95×103 m2. The transmitter has a transmission

power of Pt = 200 kW, gain of Gt = 22 dBi and the receiver has a receiver

gain of Gr = 10 dBi, the system losses are of Ls = 4 dB and the noise power

is of σ2n = 3× 10−14 W.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Nor

mal

ized

Pow

er p

atte

rn (

dB)

-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0α = 0.6

wo

w1

w2

0.5(w1+w

2)

Figure 10.43: Beampatterns adopted for clutter analysis simulation.

Figure 10.44: Original Capon spectrum of the clutter plus noise environment.

Fig. 10.44 depicts the Capon spectrum of the simulated scenario using

the clutter covariance matrix generated using only the original beampattern

depicted in blue (wo) in Fig. 10.43.

Fig. 10.45 depicts the Capon spectrum of the simulated scenario using

the mean clutter covariance matrix according to the beampatterns depicted in

Fig. 10.43 used for sidelobe modulation.

Fig. 10.46 depicts the cut of the Capon spectrum depicted in Figs 10.45

and 10.44 for u = −0.6, which is the direction of the communication receiver,

where we expect the largest variation between the original beampattern and

the mean beampattern.

Fig. 10.47 depicts the cut of the Capon spectrum depicted in Figs 10.45

and 10.44 for a fixed Doppler frequency of fD = 0 Hz, which is the Doppler of

the main clutter (where the clutter is more intense).

We can note from Figs. 10.44, 10.45, 10.46 and 10.47 that the phe-

DBD



Figure 10.45: Capon spectrum of the clutter plus noise environment usingsidelobe modulation.

Doppler frequency-150 -100 -50 0 50 100 150

Pow

er (

dBW

)

-158

-157

-156

-155

-154

-153

-152

-151

-150

-149Capon Spectrum for u = -0.6

Rc original

Rc mean

Figure 10.46: Cut of the Capon spectrum depicted in Figs 10.45 and 10.44 foru = −0.6.

nomenon of Doppler spread using the space-time MVDR filter is absent. The

sidelobe modulation does not cause any degradation to the space-time MVDR

filter.

DBD



u-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

Pow

er (

dBW

)

-160

-155

-150

-145

-140

-135

-130Capon Spectrum for fD = 0 Hz

Rc original

Rc mean

Figure 10.47: Cut of the Capon spectrum depicted in Figs 10.45 and 10.44 forfD = 0 Hz.

10.9 Conclusion

We have proposed a different formulation of the dual-function radar-

communications problem. This original formulation allowed us to develop new

radar-embedded sidelobe modulation methods that have closed form expres-

sions and are applicable to both amplitude and phase sidelobe modulation.

Our design uses a constrained optimization problem for generating

transmit beampatterns, that match satisfactorily a given transmit profile

(with high fidelity adjustment at the mainlobe), where each beampattern

embeds a different symbol towards the communication receiver direction. In

this thesis, we have also proposed an alternative way of dealing with the

mainlobe adjustment requirement. The alternative proposed method is based

on eigenvalue and point constraints.

Studying the DFRC topic, we have realized that the DFRC methods

based in varying the transmit beampattern parameters in specific directions

may provide operational bit error rates only towards that directions, what

is interesting for secure communications purposes. This advantage, though,

becomes a major disadvantage if the real receiver position doesn’t match

exactly the predefined one. This situation can easily arise if the communication

receiver is moving relative to the radar. The methods proposed in the literature

so far haven’t addressed this important issue.

Therefore, we have proposed a modification of our problem formulation

by adding a derivative null constraint. The proposed methods considering the

derivative null constraint still have closed form expressions. The new constraint

copes with this problem in a simple and yet effective manner. The derivative

null constraint imposes the beampattern to sustain the symbol’s complex

DBD



value over a small angular region, making the communication robust against

small angular deviations. The addition of this other constraint does’t lead to

significant increase of the computational complexity and solves an important

problem of the DFRC system.

In this thesis, we have also considered the important topic of adaptation

in real-time applications when the communication receiver is moving relative

to the radar. When we are dealing with dynamic scenarios, it is not feasible to

keep solving, in real time, sophisticated optimization problems, or even convex

problems by interior point techniques (as proposed in the DFRC literature

so far). Therefore, we simplified our two proposed methods and derived low-

complexity expressions for updating the beampatterns for following a moving

communication receiver platform. The derivative null constraint itself already

provides some flexibility in terms of update time, since the communication

service is not so rapidly degraded with the movement. Thus, our proposed

methods are well suited for online real-time processing.

We could also note that our simple, closed form solutions generates prac-

tically the same beampatterns, for both amplitude and phase modulation, as

the methods of [5] and [4], which use interior point technique for computing

the solutions to their problem formulation. The advantages of our proposed

methods are that they have low-complexity closed form expressions which can

be updated for following a moving communication platform with minimum

computational burden. These advantages by their own are enough for prefer-

ring our proposed methods. But we go even further and face the robustness

issue!

Referring to phase modulation, we proposed a new non-coherent sig-

nalling strategy within the DFRC context. In the literature so far, we can only

find non-coherent phase modulation based on the simultaneous transmission

of the symbol and the phase reference (multi-waveform Differential PSK-based

methods). These methods require more sophisticated technology than the ne-

cessary to change only the transmit beampattern of only one array of sensors.

To cope with this issue, we proposed a signaling strategy that is suitable for

simpler radars, that have only one transmit array. We have also derived the

analytical bit error rate expression for the proposed phase modulation sig-

nalling strategy when two arbitrary symbols are transmitted.

DBD


11Conclusion

In this thesis, we presented our accomplished work in two main top-

ics: reduced-rank beamforming and space-time processing and radar-embedded

communications. This chapter summarizes our results and highlights possibil-

ities of research directions.

Our work in rank reduced array signal processing produced interesting

contributions. We have explored the application of the JIDS in two different

areas: beamforming and space-time radar processing. The JIDS has never been

tested within these contexts before and showed impressive results! By exploring

the specificities of the beamforming environment, we were able to propose

simplifications that led to a significant computational complexity reduction.

A further work could be the investigation of the specificities of the space-

time environment to see if it is possible to come up with simplifications for

this application as well. There are still many open questions relative to the

JIDS, like, for example, how to set the decimation factor F for a good tradeoff

between complexity and performance and theoretical performance bounds as

well.

Our work in the topic of DFRC sidelobe modulation has led to many

contributions to the field. Those contributions are explained in Section 10.9.

We summarize them here focusing on further researches possibilities:

– We have proposed two original robust radar-embedded sidelobe

phase/amplitude modulation methods which have simple closed form

equations.

– To the best of our knowledge, the need for robustness was not noticed

by any other researcher in the field. We have not only alerted about this

necessity as we have also proposed a simple and effective way of turning

our methods robust against small angular errors in the relative position

between the radar and the communication receiver.

– To the best of our knowledge, concerning the need of dealing with real-

time issues, our work is the only one in the field that, besides pointing

the potential dynamic characteristic of the DFRC application, effectively

proposes a practical way for dealing with it. As our methods have

DBD


Chapter 11. Conclusion 191

low-computational complexity closed form equations, we were able to

derive practical update equations for pursuing a moving communication

platform.

– We have also proposed a simple signalling strategy for sidelobe phase

modulation and derived an analytical BER expression for it when two

arbitrary symbols are transmitted, which is a common situation within

the DFRC context.

– We have also started to work with the proposed recursive version of one

of our proposed methods and we have noticed the potential of the stop

criterium. A future work could be the investigation of a stop criterium

that is directly related to the radar operation. A first action could be

keeping the sidelobes bellow the maximum permitted level and also

minimizing the mainbeam error. The feasibility of this suboptimal (in

the mean squared sense) solution is an interesting topic, as it depends

on the initial starting point.

– Another issue that offers a vast field of investigation is the analysis of

the effects of sidelobe modulation in clutter mitigation techniques. We

have presented the simple case where there is no relative movement and

the clutter mitigation technique is the space-time MVDR filter. We have

used the true mean covariance matrix in our experiments. Further works

would involve the consideration of relative movement, simulations using

estimated covariance matrices from different sample supports and also

other clutter mitigation techniques such as Doppler filtering for example.

Finally, in this thesis we explained the accomplished studies and we

pointed out directions for further researches in the area of array signal

processing.

DBD


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DBD


AAppendix

In this appendix we derive the analytical BER expression for the DBPSK

modulation when two arbitrary symbols are transmitted.

The DBPSK receiver uses the following decision rule

ℜ{z[i]} ≷01 0, (A.1)

where z[i] is defined in (9.86)

z[i] = Er|ci(u)||ci−1(u)|ej(φi−φi−1) +N [i].

The decision regions of the DBPSK case is illustrated in Fig. A.1.

One of two cases may happen if a bit “0” is transmitted, i.e. the null

hypothesis, H0, is true. First case, denoted C0,0: C0,0 implies that ci(u) and ci−1(u)

in (9.86) are equal to C0. Second case, denoted C1,1: C1,1 implies that ci(u) and

ci−1(u) in (9.86) are equal to C1. The probability of occurring an error given

hypothesis H0 is

P (e|H0) = P (C0,0)P (e|H0, C0,0) + P (C1,1)P (e|H0, C1,1). (A.2)

Assuming that P (C0,0) = P (C1,1) = 1/2 we have that

P (e|H0) =1

2[P (e|H0, C0,0) + P (e|H0, C1,1)] . (A.3)

Considering the decision rule (A.1) we can rewrite (A.3) as

P (e|H0) =1

2[P (ℜ{z[i]} < 0|H0, C0,0) + P (ℜ{z[i]} < 0|H0, C1,1)] . (A.4)

Given that a bit “1” is transmitted, i.e. the hypothesis H1 is true, the

probability of occurring an error is

P (e|H1) =1

2[P (e|H1, C1,0) + P (e|H1, C0,1)] , (A.5)

=1

2[P (ℜ{z[i]} > 0|H1, C1,0) + P (ℜ{z[i]} > 0|H1, C0,1)] ,

(A.6)

where case C1,0 implies that ci(u) = C1 and ci−1(u) = C0 in (9.86) and case C0,1

implies that ci(u) = C0 and ci−1(u) = C1 in (9.86).

DBD


Appendix A. Appendix 200

Figure A.1: Illustration of a DBPSK modulation and its decision regions.

The error probability is given by

P (e) = P (H0)P (e|H0) + P (H1)P (e|H1). (A.7)

Since bits “0” and “1” are equiprobable we have

P (e) =1

2[P (e|H0) + P (e|H1)] , (A.8)

=1

2

{1

2[P (e|H0, C0,0) + P (e|H0, C1,1)] +

1

2[P (e|H1, C1,0) + P (e|H1, C0,1)]

}

.

(A.9)

Noting that

ℜ{r[i]r∗[i− 1]} =

∣∣∣∣

r[i] + r[i− 1]

2

∣∣∣∣

2

−∣∣∣∣

r[i]− r[i− 1]

2

∣∣∣∣

2

(A.10)

and defining

w1 ,r[i] + r[i− 1]

2and (A.11)

w2 ,r[i]− r[i− 1]

2, (A.12)

thus, ℜ{z[i]} < 0 ⇒ |w2| > |w1| and ℜ{z[i]} > 0 ⇒ |w1| > |w2|. Therefore

P (e|H0, C0,0) = P (|w2| > |w1||H0, C0,0), (A.13)

P (e|H0, C1,1) = P (|w2| > |w1||H0, C1,1), (A.14)

P (e|H1, C1,0) = P (|w1| > |w2||H1, C1,0), (A.15)

P (e|H1, C0,1) = P (|w1| > |w2||H1, C0,1). (A.16)

DBD



Let us observe (A.13). Note that

P (|w2| > |w1||H0, C0,0) =

∫ +∞

0

P (|w2| > |w1|, |w1| = α|H0, C0,0)dα, (A.17)

=

∫ +∞

0

P (|w2| > |w1|||w1| = α,H0, C0,0)p|w1||H0,C0,0(α)dα.

(A.18)

But as P (|w2| > |w1||H0, C0,0) is conditionally independent of |w1| assuming any

value, we can write (A.18) as

P (|w2| > |w1||H0, C0,0) =

∫ +∞

0

P (|w2| > α|H0, C0,0)p|w1||H0,C0,0(α)dα.

(A.19)

We can now write

P (|w2| > |w1||H0, C0,0) =

∫ +∞

0

(∫ ∞

α

p|w2||H0,C0,0(x)dx

)

p|w1||H0,C0,0(α)dα,

(A.20)

P (|w2| > |w1||H0, C1,1) =

∫ +∞

0

(∫ ∞

α

p|w2||H0,C1,1(x)dx

)

p|w1||H0,C1,1(α)dα,

(A.21)

P (|w1| > |w2||H1, C1,0) =

∫ +∞

0

(∫ ∞

α

p|w1||H1,C1,0(x)dx

)

p|w2||H1,C1,0(α)dα,

(A.22)

P (|w1| > |w2||H1, C0,1) =

∫ +∞

0

(∫ ∞

α

p|w1||H1,C0,1(x)dx

)

p|w2||H1,C0,1(α)dα.

(A.23)

But

w1 =1

2(√

Er|ci(u)|ejφi + ni +√

Er|ci−1(u)|ejφi−1 + ni−1), (A.24)

which is a complex Gaussian variable with mean µ1 =√Er(|ci(u)|ejφi +

|ci−1(u)|ejφi−1)/2 and variance given by σ2n/2 = N0/2, and

w2 =1

2(√

Er|ci(u)|ejφi + ni −√

Er|ci−1(u)|ejφi−1 − ni−1), (A.25)

which is a complex Gaussian variable with mean µ2 =√Er(|ci(u)|ejφi −

|ci−1(u)|ejφi−1)/2 and variance given by σ2n/2 = N0/2. The random variables

w1 and w2 can be shown to be independent. If we write,

wi = wi,R + jwi,I , i = 1, 2, (A.26)

DBD



then variables wi,R and wi,I , i = 1, 2, have variance given by σ2w = σ2

n/4 = N0/4.

GivenH0 and C0,0, we have that ci(u) = ci−1(u) = C0, thus |µ1| =√Er|C0|

and |w1| follows the Rician distribution,

p|w1||H0,C0,0(x) =

x

σ2w

e−x2+|µ1|

2

2σ2w I0

(x|µ1|σ2w

)

, x ≥ 0, (A.27)

where I0(·) is the modified Bessel function of the first kind and zero-th order,

|µ2| = 0 and |w2| follows the Rayleigh distribution,

p|w2||H0,C0,0(x) =

x

σ2w

e− x2

2σ2w , x ≥ 0. (A.28)

GivenH0 and C1,1, we have that ci(u) = ci−1(u) = C1, thus |µ1| =√Er|C1|

and |w1| follows the Rician distribution,

p|w1||H0,C1,1(x) =x

σ2w

e−x2+|µ1|

2

2σ2w I0

(x|µ1|σ2w

)

, x ≥ 0, (A.29)

|µ2| = 0 and |w2| follows the Rayleigh distribution, p|w2||H0,C1,1,

p|w2||H0,C1,1(x) =x

σ2w

e− x2

2σ2w , x ≥ 0. (A.30)

Given H1 and C1,0, we have that ci(u) = C1 and ci−1(u) = C0, thus

|µ1| =√Er|C1 + C0|/2 and |w1| follows the Rician distribution,

p|w1||H1,C1,0(x) =x

σ2w

e−x2+|µ1|

2

2σ2w I0

(x|µ1|σ2w

)

, x ≥ 0, (A.31)

|µ2| =√Er|C1 − C0|/2 and |w2| follows the Rician distribution,

p|w2||H1,C1,0(x) =

x

σ2w

e−x2+|µ2|

2

2σ2w I0

(x|µ2|σ2w

)

, x ≥ 0. (A.32)

Given H1 and C0,1, we have that ci(u) = C0 and ci−1(u) = C1, thus

|µ1| =√Er|C0 + C1|/2 and |w1| follows the Rician distribution,

p|w1||H1,C0,1(x) =

x

σ2w

e−x2+|µ1|

2

2σ2w I0

(x|µ1|σ2w

)

, x ≥ 0, (A.33)

|µ2| =√Er|C0 − C1|/2 and |w2| follows the Rician distribution,

p|w2||H1,C0,1(x) =x

σ2w

e−x2+|µ2|

2

2σ2w I0

(x|µ2|σ2w

)

, x ≥ 0. (A.34)

Now we can compute the integrals in (A.20), (A.21), (A.22) and (A.23).

Substituting (A.29) and (A.30) into (A.20), we have that P (e|H0, C0,0) is

given by

P (e|H0, C0,0) =

∫ +∞

0

(∫ ∞

α

x

σ2w

e− x2

2σ2w dx

)α

σ2w

e−α2+|µ1|

2

2σ2w I0

(α|µ1|σ2w

)

dα. (A.35)

DBD



Since ∫ ∞

α

x

σ2w

e− x2

2σ2w dx = e−α2

2σ2w , (A.36)

we obtain

P (e|H0, C0,0) =

∫ +∞

0

α

σ2w

e− 2α2+|µ1|

2

2σ2w I0

(α|µ1|σ2w

)

dα. (A.37)

If we substitute u =√2α and du =

√2dα into (A.37) we obtain

P (e|H0, C0,0) =

∫ +∞

0

u√2σ2

w

e−u2+|µ1|

2

2σ2w I0

(u|µ1|√2σ2

w

)

du, (A.38)

=e− |µ1|

2

4σ2w

2

∫ +∞

0

u

σ2w

e−u2+|µ1|

2/2

2σ2w I0

(

u|µ1|/√2

σ2w

)

du.

(A.39)

Noting that we are integrating a Rician distribution from 0 to +∞ in (A.39) the

solution of (A.37) is given by

P (e|H0, C0,0) =1

2e−Er|C0|

2

4σ2w . (A.40)


given by

P (e|H0, C1,1) =

∫ +∞

0

(∫ ∞

α

x

σ2w

e− x2

2σ2w dx

)α

σ2w

e−α2+|µ1|

2

2σ2w I0

(α|µ1|σ2w

)

dα (A.41)

The solution of (A.41) is equal to the solution of (A.35), except that |µ1|2 for thecase C1,1 is given by Er|C1|2. Therefore

P (e|H0, C1,1) =1

2e−Er|C1|

2

4σ2w . (A.42)


given by

P (e|H1, C1,0) =

∫ +∞

0

(∫ ∞

α

x

σ2w

e−x2+|µ1|

2

2σ2w I0

(x|µ1|σ2w

)

dx

)α

σ2w

e−α2+|µ2|

2

2σ2w I0

(α|µ2|σ2w

)

dα

(A.43)Since the Marcum Q function of a and b, QM (a, b), is defined as

QM(a, b) =

∫ +∞

b

xe−x2+a2

2 I0(ax)dx, (A.44)

equation (A.43) can be written as

P (e|H1, C1,0) =

∫ +∞

0

QM

( |µ1|σw

,α

σw

)α

σ2w

e−α2+|µ2|

2

2σ2w I0

(α|µ2|σ2w

)

dα,

(A.45)

DBD



Substituting the values of |µ1| and |µ2| for case C1,0, which are |µ1| =√Er|C1 +

C0|/2 and |µ2| =√Er|C1 − C0|/2, we have

P (e|H1, C1,0) =

∫ +∞

0

QM

( √Er|C1+C0|

2

σw,α

σw

)

α

σ2w

e−α2+

Er|C1−C0|2

42σ2w I0

(

α√Er |C1−C0|

2

σ2w

)

dα.

(A.46)

Equation (A.46) is the analytical value of P (e|H1, C1,0), but the integral in (A.46)

has to be computed numerically.


given by

P (e|H1, C0,1) =

∫ +∞

0

(∫ ∞

α

x

σ2w

e−x2+|µ1|

2

2σ2w I0

(x|µ1|σ2w

)

dx

)α

σ2w

e−α2+|µ2|

2

2σ2w I0

(α|µ2|σ2w

)

dα

(A.47)The solution of (A.47) is equal to the solution of (A.46), except for the values of

|µ1| and |µ2|, which are |µ1| =√Er|C0+C1|/2 and |µ2| =

√Er|C0−C1|/2. But

note that |µ1| and |µ2| given hypothesis H1 and case C0,1 are equal to the values

of |µ1| and |µ2| given hypothesis H1 and case C1,0. Therefore

P (e|H1, C0,1) = P (e|H1, C1,0), (A.48)

which is given in (A.46).

Substituting (A.34), (A.42), (A.46) and (A.48) into (A.8), we have that

the error probability, P (e), is given by

P (e) =1

8e−Er|C0|

2

4σ2w +1

8e−Er |C1|

2

4σ2w

+1

2

∫ +∞

0

QM

( √Er|C1+C0|

2

σw,α

σw

)

α

σ2w

e−α2+

Er |C1−C0|2

42σ2w I0

(

α√Er|C1−C0|

2

σ2w

)

dα.

(A.49)

DBD


Aline de Oliveira Ferreira Contributions to Array Signal ... · To Dr. Marcello Campos, whose...

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